Quantitative modeling of multi-signal quorum sensing maps environment to bacterial regulatory responses

Stephen Thomas^1*, Ayatollah S. El-Zayat², James Gurney³, Jennifer Rattray^1,#, Sam P. Brown^1*

1. School of Biological Sciences, Georgia Institute of Technology, Atlanta, Georgia, USA.

2. Department of Microbiology, Faculty of Agriculture, Cairo University, Giza, Egypt.

3. Department of Biology, College of Arts and Sciences, Georgia State University, Atlanta, Georgia, USA.

#. Current address: Sandia National Laboratories, Albuquerque, New Mexico, USA.

* [email protected], [email protected]

Abstract: Bacterial quorum sensing is often mediated by multiple signaling systems that interact with each other. The quorum sensing systems of Pseudomonas aeruginosa, for example, are considered hierarchical, with the las system acting as a master regulator. By experimentally controlling the concentration of auto-inducer signals in a signal deficient strain (PAO1ΔlasIΔrhlI), we show that the two primary quorum sensing systems—las and rhl—act reciprocally rather than hierarchically. Just as the las system’s 3‑oxo‑C₁₂‑HSL can induce increased expression of rhlI, the rhl system’s C₄‑HSL increases the expression level of lasI. We develop a mathematical model to quantify relationships both within and between the las and rhl quorum sensing systems and the downstream genes they influence. The results show that not only do the systems interact in a reciprocal manner, but they do so asymmetrically, cooperatively, and nonlinearly, with the combination of C₄‑HSL and 3‑oxo‑C₁₂‑HSL increasing expression level far more than the sum of their individual effects. We next extend our parameterized mathematical model to generate quantitative predictions on how a QS-controlled effector gene (lasB) responds to changes in wildtype bacterial stationary phase density and find close quantitative agreement with an independent dataset. Finally, we use our parameterized model to assess how changes in multi-signal interactions modulate functional responses to variation in social (population density) and physical (mass transfer) environment and demonstrate that a reciprocal architecture is more responsive to density and more robust to mass transfer than a strict hierarchy.

Significance Statement

Bacteria often sense and respond to their environment via quorum sensing. While we understand the core components of signal production and response in great molecular detail, we face a fundamental gap in our understanding of how these components combine to quantitatively map environmental properties onto behavioral responses. To address this gap, we develop a dynamical-systems approach, using controlled experimental data to parameterize and test a mathematical model of multi-signal QS, using Pseudomonas aeruginosa as a model system. Our approach provides a general and flexible methodology to quantify the multi-signal “wiring diagrams” of QS bacteria and functionally link these wiring diagrams to QS-controlled responses to environmental variation.

Introduction

Within many bacterial species, cells communicate with each other by exchanging diffusible signal molecules (Papenfort and Bassler 2016; Whiteley, Diggle, and Greenberg 2017). This mechanism, known as quorum sensing (QS), has been well-studied at the level of specific molecular interactions. We now understand how those molecular interactions shape the creation of and response to signal molecules in model organisms such as Pseudomonas aeruginosa (Pearson, Pesci, and Iglewski 1997). We have identified downstream effector genes such as virulence factors whose production depends on QS signals (Chugani et al. 2001; Kiratisin, Tucker, and Passador 2002), and we have recognized that many species possess multiple QS circuits (Papenfort and Bassler 2016). Despite this knowledge, we face gaps in our understanding of how quorum sensing influences bacterial behavior. How does QS quantitatively guide bacterial actions in response to defined environmental conditions? What benefits do multiple QS circuits provide? And ultimately, how does QS contribute to bacterial fitness? Answering these questions requires an understanding of quorum sensing at the dynamical systems level as well as the molecular level.

Quorum sensing relies on several components interacting in a dynamical system (Popat et al. 2015; Pérez-Velázquez and Hense 2018). Individual cells synthesize small molecules called signals or inducers. These diffuse or are actively transported between the intracellular and extracellular environments (Pearson, Van Delden, and Iglewski 1999). Within cells, signal molecules bind to receptor proteins that can then serve as transcription factors (Bottomley et al. 2007). As signal concentration grows, genes activated by these transcription factors trigger a change in the cell's behavior (Whiteley, Lee, and Greenberg 1999). Those molecules related to a particular signal—the signal synthase, the signal molecule, and the cognate receptor—form a quorum sensing system. Some bacterial species have multiple QS systems, the opportunistic pathogen Pseudomonas aeruginosa among them. Among its multiple systems, las and rhl acyl-homoserine lactone (AHL) signaling systems have been especially well studied (Pesci et al. 1997; Lee and Zhang 2015). The las system includes the LasI signal synthase, N-(3-oxododecanoyl)-l-homoserine lactone (3‑oxo‑C₁₂‑HSL) signal, and LasR receptor. The corresponding components of the rhl system are RhlI, N-butyryl-homoserine lactone (C₄‑HSL), and RhlR. Schuster and Greenberg (2007) estimate that these two systems control expression of as much as 10% of the bacterial genome.

P. aeruginosa provides a model for understanding interactions between multiple QS systems. How does the behavior of one system, determined by the concentration of signal it produces, affect the behavior of a different system? How does expression of one system’s synthase or receptor respond to the concentration of another systems signal? We classify possible multi-system architectures into three broad patterns shown in Fig 1. Independent systems (Fig 1A) have no influence on each other. Hierarchical systems (Fig 1B) have a relationship but only in one direction; one QS system exerts an influence on the other, but the second has no effect on the first. Finally, reciprocal systems (Fig 1C) each exert influence on the other. At this level we do not make any assumptions concerning the underlying mechanism(s) of the inter-system effects. For example, the signal of one system may bind directly to the receptor of the other; alternatively, the signal/receptor complex of one system may act as a transcriptional regulator of components in the second system. In both cases we simply denote the first system as influencing the second and focus on quantifying the degree of influence, regardless of mechanistic pathway.

Figure 1: architectures

Fig 1. The P. aeruginosa QS regulatory network is typically viewed as a hierarchy with the las system on top. In general, the regulatory network of two QS systems may be organized in three architectures: (A) Independent, in which the signal of each has no influence on the expression of synthase or receptor in the other. (B) Hierarchical, where one system’s signal influences expression of the other’s components but without reciprocation. And (C) reciprocal, where both systems’ signals influence the others’ components. The consensus of the review literature for P. aeruginosa (D) as summarized in 17 review papers published since 1996 (Tables A and B in S1 Text) suggests a hierarchical architecture for the las and rhl systems.

In the case of las and rhl, independent, isolated operation was eliminated as early as 1996 when Latifi et al. (1996) used lacZ transcriptional fusions to show that the combination of LasR and 3‑oxo‑C₁₂‑HSL controls expression of rhlR, demonstrating that the las system influences the rhl system. These and other results have led many researchers to view las and rhl as a hierarchy, with the las system serving as the master QS system controlling both its own activation and that of the rhl system (Fig 1D). Notably, in a strict hierarchy, the rhl system exerts no influence on the las system. We confirm this consensus perspective via a structured literature search of review articles (Tables A and B in S1 Text). All papers show the las system affecting the rhl system, but none identify the las synthase or receptor gene as a target of the rhl system. Despite its prevalence in the review literature, experimental studies have flagged limitations in the hierarchical model. For example, several studies demonstrate that the rhl system can self-activate and drive expression of QS-controlled virulence factors, even when the “top of the hierarchy” lasR gene is experimentally deleted (Dekimpe and Déziel 2009) or functionally compromised during experimental or clinical evolution (Kostylev et al. 2019). These results indicate a potential genetic transition to an independent signal architecture (Fig 1A) with one functional signal system (rhl) and raise the broader question of how changes in multi-signal wiring impact QS functionality. We approach this question by using experimental data from the model P. aeruginosa strain PAO1 to test all three wiring models in Figs 1A-1C. If lasI or lasR respond to the rhl signal, then the strict hierarchical view is potentially missing an important factor affecting the overall system response, a factor potentially captured by a reciprocal wiring diagram (Fig 1C).

Our experiments explicitly examine the influence of both QS systems on each other, and the resulting data reveal three key results. First, the traditional, strict hierarchical view of las and rhl is incomplete. Our results confirm that las can exert control over the rhl system, but we also observe the converse: rhl substantially influences the las system, specifically expression of lasI. Second, we use our data to parameterize a new mathematical model of multi-signal regulatory dynamics, producing quantitative estimates on how signal modules influence each other and the expression of downstream effector genes. Finally, we embed our multi-signal model in an explicit environmental context to produce and test quantitative predictions on how QS-controlled behaviors change with changing environmental conditions. Specifically, we show that, compared to a strict hierarchy, the reciprocity we observe provides more sensitivity to population density and more robustness to interfering physical environmental variation.

Results

To uncover interactions between the las and rhl systems, we begin by experimentally assessing QS gene expression in an AHL null strain (PAO1ΔlasIΔrhlI) exposed to defined, exogenous concentrations of the signal molecules 3‑oxo‑C₁₂‑HSL (las system) and C₄‑HSL (rhl system). We use bioluminescence (lux) transcriptional fusion reporters for lasI and rhlI to estimate expression levels of the respective signal synthase genes. We then develop mathematical models to quantify the effects of each system on the other and their consequent responses to environmental variation. In a final experimental step, we test our predicted QS responses to differing bacterial densities in a strain with intact regulatory feedbacks governing endogenous signal production.

The las and rhl Systems Interact Reciprocally

We first evaluate quorum sensing behavior under the influence of a single signal. We establish a baseline expression level by measuring reporter luminescence with no signal present. We then observe the increase in luminescence as exogenously controlled signal concentration increases. We chose signal concentrations to reflect experimentally observed concentrations typically ranging from 0.1 to 5 μM (Juříková et al. 2023). The ratio of luminescence with signal to luminescence with no signal represents the fold-change in expression induced by the defined signal concentration. Figs 2A and 2D show the results for 3‑oxo‑C₁₂‑HSL. As expected, expression of both lasI and rhlI genes increases as signal concentration increases. The availability of the las signal molecule influences the expression of rhlI as well as lasI, and, therefore, the las system affects the rhl system.

Figure 2: observations

Fig 2. Both the las signal 3‑oxo‑C₁₂‑HSL and the rhl signal C₄‑HSL increase the expression of lasI and rhlI in a signal null PAO1. (A-C) lasI expression level as a function of defined concentrations of 3‑oxo‑C₁₂‑HSL alone (A), C₄‑HSL alone (B), and both signals together (C). (D-F) rhlI expression level under the same conditions. All plots show fold-change in RLU/OD (relative light units per optical density) values compared to baseline with no exogenous signals in NPAO1∆lasI∆rhlI cultures. Genomic reporter fusions lasI:luxCDABE and rhlI:luxCDABE were used to generate luminescence. Points are individual observations within the time window of peak expression; lines show predictions from the Michaelis-Menten model of Eq A in S1 Text parameterized according to Table C in S1 Text. Figs A and B in S1 Text show the underlying expression data for the entire time course of the experiments. Additional observations at lower signal concentrations were used to validate the model parameters and showed strong agreement (R² = 0.82) as detailed in Fig D in S1 Text. (The data underlying this Figure and the code used to analyze it can be found in https://doi.org/10.5281/zenodo.15808353.)

While we find no surprises with 3‑oxo‑C₁₂‑HSL, our experiments with C₄‑HSL challenge the conventional hierarchical view. Figs 2B and 2E show those results: expression of lasI and rhlI increases with higher C₄‑HSL concentration. The response of lasI (Fig 2B) does not correspond to a strict hierarchy with las as the master. Here we find that the rhl system affects the las system.

To quantify the impact of each signal alone, we model gene expression using Michaelis-Menten kinetics under quasi-steady state assumptions (Methods), allowing us to estimate both the maximal fold-change under signal activation and sensitivity to signal (Table C in S1 Text). Our model fits (Figs 2A, 2B, 2D, 2E, and Fig D in S1 Text) indicate that while the las and rhl systems have reciprocal impacts, those impacts are not symmetrical. The las signal 3‑oxo‑C₁₂‑HSL has a substantially greater influence on gene expression than C₄‑HSL. In both cases the potential fold-change from 3‑oxo‑C₁₂‑HSL is approximately six times greater than the potential fold-change from C₄‑HSL. Both lasI and rhlI are also more sensitive to 3‑oxo‑C₁₂‑HSL than to the C₄‑HSL as the concentrations required to reach half of maximal expression are roughly 4 times and 30 times higher for the latter.

Figs 2A, 2B, 2D, and 2E consider the effects of each signal in isolation, but wildtype cells with functioning synthase genes can produce both signals. To understand environments where both signals are present, we use controlled concentrations of both signals in combination. Figs 2C and 2F present those results in the form of heat maps. The qualitative responses of both genes are similar: raising the concentration of either signal increases expression regardless of the concentration of the other signal. As with our observations of C₄‑HSL alone, these results demonstrate again that the rhl system (via C₄‑HSL) affects the las system (lasI expression).

Quantifying the las and rhl Interactions Reveals Non-Linear and Synergistic Effects

Having established a simple Michaelis-Menten model for each signal in isolation (Figs 2A, 2B, 2D, and 2E), we next consider whether that model is sufficient to explain the effect of the signals in combination (Figs 2C and 2F). Can we estimate total expression as the sum of expression induced by each signal alone? Such a response could result from two independent binding sites in the promoter regions (Buchler, Gerland, and Hwa 2003), one site for LasR/3‑oxo‑C₁₂‑HSL and a separate site for RhlR/C₄‑HSL. Figs 3A and 3B clearly show that we cannot. The maximum expression observed, shown as a “ceiling” in that figure’s panels, far exceeds the sum of the signals’ individual influence.

Figure 3: reciprocal

Fig 3. The las and rhl QS systems have a reciprocal, synergistic, and unequal relationship. (A,B) Single-signal models demonstrate that the summed effects of single signals (3‑oxo‑C₁₂‑HSL alone, red; C₄‑HSL alone, orange) cannot account for the maximal expression of lasI or rhlI. The upper green, flat surfaces in the plots indicate the maximum mean expression level measured across all combinations of signal concentrations while the lower semi-transparent surfaces mark the sum of single signal effects. The plotted points represent observed expression levels when C₄‑HSL is withheld (red) and when 3‑oxo‑C₁₂‑HSL is withheld (yellow). Lines indicate the model predictions (Eq A in S1 Text, parameters in Table C in S1 Text). (C,D) Multi-signal non-linear models capture the synergistic effects of both signals and match observed expression levels. Model estimates are shown as grid lines. Horizontal bars show the mean value of expression observed at each combination of signal concentrations. Lines extend from these mean values to the relevant grid point for clarity. The coefficient of determination (R²) for the models is 0.82 and 0.77, respectively. Figs E and F in S1 Text present more detailed comparisons between model and observations. (E) The data of Fig 2 shows that relationship of the las and rhl systems is reciprocal, and the multi-signal model quantifies the strength of those interactions. In particular, it reveals the contribution of both signals to the maximum fold-change in expression of both synthases. The charts in this panel summarize the contribution of 3‑oxo‑C₁₂‑HSL (red), C₄‑HSL (yellow), and the synergistic combination of both (orange). (The data underlying this Figure and the code used to analyze it can be found in https://doi.org/10.5281/zenodo.15808353.)

To account for the synergy between the signals, we incorporate a cooperativity term in the gene expression model. Note that the cooperativity term is a multiplication of signals, and it alone cannot explain the full response, as the product is necessarily zero when any signal is absent. This term accounts for any non-additive interaction, for example the ability of one bound transcription factor to recruit the binding of a second transcription factor (Kaplan et al. 2008). Eq 1 shows the result. Each gene has a basal expression level, amplification from each signal alone, and additional amplification from each pair-wise combination of signals. The interaction from these pair-wise combinations captures the cooperative enhancement from the combined signals.

$$E_i(\mathbf{S}) \ \ = \ \ \alpha_{i,0} \ \ \ + \ \ \ \sum_{j=1}^{\mathrm{N_S}}\alpha_{i,j}\frac{[S_j]}{[S_j] + K_{i,j}} \ \ \ + \ \ \\ \sum_{j=1}^{\mathrm{N_S}-1}\sum_{j' = j+1}^{\mathrm{N_S}} \alpha_{i,j,j'}\frac{[S_j][S_{j'}]}{([S_j] + K_{Qi,j,j'})([S_{j'}] + K_{Qi,j',j})} \qquad{(1)}$$

For both lasI and rhlI we again minimize the sum of squared errors to estimate parameter values (non-linear regression using the Gauss-Newton algorithm). The resulting multi-signal models in Table D in S1 Text have R² values of 0.82 and 0.77. Figs 3C and 3D compare the model estimates with observations. For both genes, the model captures the effect of each signal in isolation and also both signals in combination.

The parameter estimates in Table D in S1 Text quantify the relative effect of individual and combined signals. The maximum expression induced by both signals nearly doubles compared to the maximum expression induced by any signal alone. Fig 3E summarizes the model parameters graphically and revises the consensus view of Fig 1D—the rhl system does influence the las system—and it shows the relative magnitudes of the effects.

las and rhl Synergy Also Shapes Control of Quorum Sensing Effector Genes

Our results, summarized in Fig 3, establish that both AHL signals influence the expression levels of both synthase genes in a synergistic manner. We note that the methods outlined in Figs 2–3 for the lasI and rhlI synthase genes can be applied to any QS-controlled gene of interest and can quantify dual (or more) signal control over expression levels. Here we look at lasB, a classic QS effector gene that codes for the secreted digestive enzyme LasB and is widely used as a model of QS-mediated virulence (Casilag et al. 2016; Cigana et al. 2021) and cooperation (Diggle et al. 2007; Allen et al. 2016; Sexton and Schuster 2017).

The prior literature provides a clear expectation that lasB expression will be positively influenced by both AHL signal molecules (Pearson, Pesci, and Iglewski 1997; Nouwens et al. 2003), but does not provide a quantitative partitioning of the relative importance of each signal’s contribution—alone and via synergistic effects. Using the approach outlined above, we measure luminescence of a lasB reporter in an AHL signal null strain exposed to defined, exogenous concentrations of both AHL signals, revealing maximal activation under dual-signal exposure (Figs 4A and 4B). As illustrated for lasI and rhlI (Fig 3), fitting the baseline single signal model (Eq A in S1 Text) cannot account for the maximal expression of lasB (Fig 4A), while the multi-signal model (Eq 1) can fit the data (Fig 4B). Fitting Eq 1 to the lasB data (Table E in S1 Text) supports previous conclusions that 3‑oxo‑C₁₂‑HSL has a stronger effect than C₄‑HSL on lasB expression (Anderson, Zimprich, and Rust 1999; J. B. Rattray et al. 2023). Our results add that lasB expression is dominated by the synergistic combination of both signals rather than either signal in isolation. We further note that lasB expression is more sensitive to C₄‑HSL alone than to 3‑oxo‑C₁₂‑HSL alone. (Fig 4C); the maximum increase in expression in response to C₄‑HSL is much less than to 3‑oxo‑C₁₂‑HSL, but C₄‑HSL requires a significantly lower concentration to achieve that maximum increase (Table E in S1 Text).

Figure 4: lasb

Fig 4. Expression of lasB is maximal in the presence of both C₄‑HSL and 3‑oxo‑C₁₂‑HSL. An analysis similar to Fig 3 shows that (A) a single-signal model cannot account for the maximal expression of lasB, while (B) a multi-signal model incorporating synergistic effects can match observations. Panel C summarizes qualitative differences in the relative effects of each signal on lasB response compared to lasI and rhlI. Pie charts break down the contribution to maximum fold-change in expression levels. For both lasI and rhlI, 3‑oxo‑C₁₂‑HSL alone is responsible for slightly more than half of their maximum expression, while for lasB 3‑oxo‑C₁₂‑HSL alone accounts for less than a quarter. The combined effect of both 3‑oxo‑C₁₂‑HSL and C₄‑HSL together dominates lasB expression. The half concentration values also show a qualitative difference. While lasI and rhl are nearly equally sensitive to 3‑oxo‑C₁₂‑HSL and C₄‑HSL, lasB is substantially more sensitive to C₄‑HSL than to 3‑oxo‑C₁₂‑HSL. Fig C in S1 Text shows underlying expression data for full time course of experiments; Table E in S1 Text lists the parameter values, and Fig G in S1 Text provides a detailed comparison of model predictions and observations. (The data underlying this Figure and the code used to analyze it can be found in https://doi.org/10.5281/zenodo.15808353.)

Mathematical Models Incorporating Multi-signal Interactions Predict Wildtype Quorum Sensing Response to Environmental Variation

Our parameterized gene expression model (Eq 1) predicts gene i’s expression E_i as a function of the AHL signal environment S (Figs 3C, 3D, 4B, and Fig G in S1 Text), which leaves open the question of how the signal environment S relates to underlying dimensions of environmental variation—the hypothesized sensing targets of quorum sensing (population density (Fuqua, Winans, and Greenberg 1994), diffusion (Redfield 2002), efficiency (Hense et al. 2007), containment (Boedicker, Vincent, and Ismagilov 2009), genotypes (Eldar 2011; Allen et al. 2016), combinations (Cornforth et al. 2014), etc.)

To connect our gene expression model to critical environmental dimensions of bacterial population density and mass transfer (e.g. diffusion or advective flow), we build on previous models of extracellular signal dynamics (James et al. 2000; Dockery and Keener 2001; Ward et al. 2001; Brown 2013; Cornforth et al. 2014). While our earlier experimental and modeling analyses (Figs 2-4 and Eq 1) treated signal concentration as an experimentally fixed exogenous factor, we now treat signal concentration as a dynamical variable that emerges from the interplay of regulatory feedbacks (both within- and among signal modules) and environmental conditions

We assume that signal concentration increases in proportion to the corresponding synthase’s expression level, multiplied by the number of cells expressing synthase, and decreases due to a constant rate of decay and removal via mass transfer. These assumptions lead to the differential equation model of Eq 2, where S_i is the concentration of signal i, E_i (S) is the expression level of the synthase for signal i (as a function of both signal concentrations, S, see Eq 1) and c_i a proportionality constant. N is the population density; 𝛿_i is the decay rate of signal i, and m is the rate of mass transfer.

$$\frac{\mathrm{d}S_i}{\mathrm{dt}} \ \ = \ \ c_i E_i(\mathbf{S})\cdot N \ \ - \ \ \delta_i \cdot S_i \ \ - \ \ m \cdot S_i \qquad{(2)}$$

This equation models the dynamics of extracellular signal concentrations in response to environmental conditions defined by density N and mass transfer rate m. While it is possible to derive analytical equilibrium solutions to Eq 2 for either independent (Figs 2A, 2B, 2D, 2E, and Eq A in S1 Text) or synergistic (Eq 1) signal activation E_i(S), the resulting solutions are cumbersome and do not yield clear insights into system behavior. (See “Analytic Solutions for Equilibrium” in S1 Text.)

To numerically solve for equilibrium signal values in Eq 2 that result from given values of population density and mass transfer, we derive parameter estimates for synergistic E_i(S) from our experimental data (Eq 1 and Table D in S1 Text). The remaining parameters, c_i and 𝜹_i, we estimate from published literature as detailed in Table E and Fig H in S1 Text. We can connect these equilibrium environmental signal values to QS-controlled effector behavior using our parameterized model of lasB expression (Fig 4 and Eq 1 as parameterized by Table E in S1 Text). Integrating these steps allows us to predict how lasB expression varies in wildtype bacteria (with intact signal synthases) as the social (N) and physical (m) environment changes.

In Figs 5A and 5B we illustrate our integrated model prediction of wildtype lasB expression as a function of change in stationary phase population density, N (solid line, Fig 5B), capturing the causal chain (Fig 5A) from environment N to signal levels S (Eq 2) and signals S to lasB expression (Eq 1 and Fig 4 and Table E in S1 Text). To test this prediction using an independent dataset, we use previously published data on wildtype lasB expression as a function of stationary phase density (J. Rattray et al. 2022). Our model prediction is in strong agreement with the independently derived data (R² = 0.91), demonstrating not only the utility of our approach but also providing theoretical support for the conclusions of the prior experimental study (J. Rattray et al. 2022): QS controlled behaviors in P. aeruginosa are not governed by a critical cell density or ‘quorum’ threshold separating off/on states, but are better described as producing graded or rheostatic responses to changes in stationary phase density.

Figure 5: environment

Fig 5. The models of Eqs 1 and 2 in combination can quantitatively predict wildtype QS-controlled lasB response to environmental variation. (A) Mathematical model schematic: Given values for population density and mass transfer rate, Eq 2 (parameterized by Table F in S1 Text) predicts equilibrium concentrations of 3‑oxo‑C₁₂‑HSL and C₄‑HSL. With those values Eq 1 (parameterized by Table E in S1 Text) can predict the resulting lasB expression level. (B) Independently parameterized model predictions compared to experimental observations. The plot shows a reaction norm (Stearns 1989) of predicted lasB expression level (solid line) as a function of population carrying capacity. The figure also shows independent experimental observations of wildtype lasB expression as a function of bacterial carrying capacity, manipulated by varying the concentration of limiting carbon (J. Rattray et al. 2022). Model predictions are in good agreement with independent experimental data (R² = 0.91), together showing that lasB responds to changes in population density and mass transfer in a graded manner. Note that the model parameters are not fitted to the data in this figure. (C-H) Heat maps of model predicted lasB expression level as a function of both mass transfer m and population density N given three quorum sensing architectures. (C-E) Architectures without rescaling; (F-H) architectures with rescaling to standardize maximum expression levels. The lines on each heat map indicate density and mass transfer values for which lasB expression is constant, either 50% of its maximum value (white) or 5% of its maximum value (black). (The data underlying this Figure and the code used to analyze it can be found in https://doi.org/10.5281/zenodo.15808353.)

Multi-Signal Architectures Govern Functional Responses to Environmental Variation

Following an initial validation of our quantitative predictions linking environmental properties to wildtype QS-controlled behaviors (Fig 5B), we now turn to a counter-factual approach to assess how changes in model architecture translate to changes in the ability to sense and respond to differences in environmental conditions. Specifically, we ask: how do reciprocal versus hierarchical versus independent architectures (Fig 1) influence QS signal and effector responses to variation in bacterial density and mass transfer? To answer that question, we recognize that Eq 1 is general enough to model hierarchical and independent architectures in addition to reciprocal architectures. Those alternatives emerge when specific ɑ parameter values are equal to zero.

Starting with our full reciprocal activation model (Table D in S1 Text), we can derive a las-controlled hierarchical model by setting ɑ_1,2 and ɑ_1,1,2 to zero (removing influences of the rhl system on the las system). To derive an independent model we additionally set ɑ_2,1 and ɑ_2,1,2 to zero (removing influences of the las system on the rhl system). These adjustments (parameters summarized in Table F in S1 Text) allow the multi-signal architecture to represent the alternate models Fig 1 depicts, but they also diminish the total weight of QS activation processes. To compensate for this diminished activation we also examine rescaled hierarchical and rescaled independent parameter sets, where the maximal activation weightings are held constant across models (parameters in Table G in S1 Text).

Using our alternate “counterfactual” models, we examine in Figs 5C-5H how equilibrium lasB expression changes over a range of population densities N and mass transfer rates m, for all three architectures. Note that in Fig 5C (reciprocal architecture) with m = 0, we recover the graded response to increasing stationary density N predicted theoretically and confirmed empirically in Fig 5B. Looking across alternate models without rescaling (Figs 5C-5E) we see that removing activating processes from rhl to las (hierarchical model) and also from las to rhl (independent model) leads unsurprisingly to a general weakening of QS maximal response. In the re-scaled models (Figs 5F-5H) we normalize maximal expression, so we can contrast functional responses to changes in density and mass transfer. Comparing these panels we can see that changes in architecture change the functional response to environmental variables. In comparison to the hierarchical or independent models, the reciprocal architecture broadens the environmental parameter space where lasB expression is elevated (areas above 5% black line, and above 50% white line) and also reduces the slope of activation contours (angles of black and white lines). The change in contour slope indicates a greater robustness to increasing mass transfer, given a reciprocal activation architecture. In Fig I in S1 Text we assess the temporal activation dynamics of lasB under the three alternate architectures and highlight that even under the rescaled model the reciprocal architecture provides the most rapid activation process (given high density and low mass transfer).

Discussion

As our knowledge of quorum sensing deepens across species, it is increasingly clear that the use of multiple QS signal molecules is the norm (e.g. E. coli (Mayer et al. 2023), B. subtilis (Auchtung, Lee, and Grossman 2006), P. polymyxa (Voichek et al. 2020), V. harveyi (Henke and Bassler 2004; Long et al. 2009), and V. cholerae (Bridges and Bassler 2019); Jung, Hawver, and Ng (2016)). While this widespread design feature has been probed extensively on a molecular mechanistic scale to produce detailed intracellular “wiring diagrams” (Papenfort and Bassler 2016; Mok, Wingreen, and Bassler 2003). we still lack a general understanding of the dynamical capacities and functional sensing role(s) of these more complex architectures. Functional analyses in Vibrio harveyi have documented how downstream behavioral responses to dual signal inputs follow a simple additive rule that potentially buffers against signal noise (Long et al. 2009; Mehta et al. 2009, 2009), but the extent to which signal modules directly impact each other has not been directly defined. In this study we use the model QS organism Pseudomonas aeruginosa to quantify the dynamical interactions within and among the las and rhl signaling sub-systems and place this dynamical behavior in the functional context of sensing both physical and social environmental variation.

Our experimental data show that the conventional las-rhl hierarchical view of QS in P. aeruginosa (Fig 1D) is incomplete. Specifically, we quantify a substantial (six-fold) increase in lasI expression compared to basal levels given exposure to the rhl signal C₄‑HSL (Fig 2B). Fitting mathematical models to data, we quantify within- and among-module interactions, and we demonstrate that the las-rhl system functions as a biased, reciprocally-activating and cooperative network (Fig 3E). To place this intracellular network in an environmental context, we extend our mathematical model to incorporate environmental properties and find close quantitative support for our model predictions using wildtype QS data (Fig 5B). Finally, we use our parameterized model to examine the impact of changing inter-module wiring and conclude that the reciprocal architecture results in QS-controlled gene expression that is more sensitive to population density and more robust in the presence of environmental interference (Figs 5C-5H).

Turning to the most common functional rationales for QS, our final parameterized model (Figs 5C-5F) rejects both the pure “density sensing” (only responsive to density) and pure “diffusion sensing” (only responsive to mass transfer) limit cases, as responsiveness to a fixed density is dependent on the level of mass transfer, and vice-versa (Hense et al. 2007; West et al. 2012; Cornforth et al. 2014). While the reciprocal architecture allows greater robustness to increasing mass transfer m, it is still the case that maximal lasB expression is concentrated in conditions of high density and low mass transfer, consistent with previous models of “efficiency sensing” (Hense et al. 2007) and combinatorial quorum sensing (Cornforth et al. 2014). Under the combinatorial QS model, information on both social and physical conditions (N and m) are encoded into the distribution of multiple signals (given differences in signal decay, 𝛿_i, and auto-regulation, ɑ_i,i, across signals), which can then be decoded by effectors given non-additive responses to multiple signal inputs. Prior work supports the assumptions of differing rates of environmental decay across AHL signals, and varied non-additive responses across effector genes, including AND-gate control of lasB (Cornforth et al. 2014), while our current analysis additionally supports specific predictions on auto-regulation rates (stronger rates of auto-regulation for the more fragile 3‑oxo‑C₁₂‑HSL signal, Table D in S1 Text).

Prior analyses of the combinatorial QS model (Cornforth et al. 2014) assumed independent signal activation and a simple threshold activation model for each signal, assumptions that are rejected by the results of the current study. To further assess the combinatorial QS model in light of our new model and data, we repeat our “counter-factual” exercise to explore distinct multi-signal wirings (Figs 5F-5H) but now applied to signal concentrations as a function of environmental conditions instead of lasB expression. Our analyses (Fig J in S1 Text) illustrate that the prediction of differential signal responses to changing environmental parameters is substantial given the independent architecture (different contour slopes in Fig J panels C and F in S1 Text), but is attenuated given the reciprocal architecture (Fig J panels A and D in S1 Text). This attenuation is not surprising given that a reciprocal architecture of activation will strengthen the coupling of activation levels between the two signals, therefore limiting the ability of two signals to report independently on distinct environmental conditions. In Fig K in S1 Text we show that despite the attenuation of differences between the two signals, the ratio of signals still changes substantially (two-fold) across the (m, N) parameter space examined, therefore maintaining an encoding of environmental conditions in the distribution of signals.

By focusing on extracellular signal concentration as the factor determining behavior, our approach accommodates multiple possible intracellular mechanisms of gene activation. And although they do not explicitly model specific mechanisms, our methods can suggest some mechanisms and rule out others. For example, the results shown in Fig 4C highlight a clear difference between the response of lasB and that of lasI or rhlI. Specifically, while all three genes require both signals for maximal expression, the contribution of C₄‑HSL alone to the expression of lasB is negligible compared to lasI or rhlI. Maximum expression of lasB only increases by a factor of 1.1 (Table E in S1 Text) with the addition of C₄‑HSL alone, in contrast to the six-fold increases for both lasI and rhlI (Table D in S1 Text). These results are consistent with cooperative promoter binding as a mechanism in the case of lasB, and they preclude cooperative promoter binding as the sole mechanism in the case of lasI and rhlI. They are also consistent with the results of Keegan et al. (2023), who identify a RhlR promoter binding domain for lasB but find no such domain for lasI. More generally, our lasI and rhlI results raise challenges for standard additive (Long et al. 2009) or multiplicative (Kaplan et al. 2008) models of gene expression as a function of multiple inputs. Neither can fully explain our results. Sauer et al. (1995) make related observations for a protein complex in Drosophila melanogaster; both of the developmental regulators BCD and HB alone induce a 6-fold increase by themselves but combine to induce a greater than 65-fold increase. These convergent results across large phylogenetic distances showcasing both additive and multiplicative regulatory components offer a tantalizing possibility that further investigations into the mechanisms of P. aeruginosa quorum sensing interactions can provide insights into more general gene regulatory network models.

Finally, we highlight that our model is not constrained to dual-signal QS, or to mutually enhancing interactions among signal systems. Fig L in S1 Text takes our two-signal las–rhl baseline and adds a third signal module (parameterized by Table H in S1 Text) that qualitatively mirrors the pqs system of P. aeruginosa (pqs is both induced by the las system and repressed by the rhl system and it, in turn, can induce the rhl system (McGrath, Wade, and Pesci 2004; Lee and Zhang 2015)). Fig L in S1 Text shows that shifts in the relative strengths of activation and repression can result in qualitative difference in behavioral outputs of the three-signal system. In sum, our methodology provides a general and flexible methodology to quantify the multi-signal “wiring diagrams” of quorum-sensing bacteria and functionally link these wiring diagrams to QS-controlled responses to environmental variation.

Methods

Literature Review

The PubMed database of the US National Institutes of Health was queried on 20 July 2021 using the query PubMed Search ("review"[Title/Abstract] OR "review"[Publication Type]) AND "quorum sensing"[Title] AND "pseudomonas aeruginosa"[Title/Abstract], resulting in 76 results with publication dates from 1996 to 2021. Papers that included a diagram of the gene transcription networks for the las and rhl quorum sensing systems were further analyzed to interpret the interactions present in those diagrams. Tables A and B in S1 Text show the results. Of the papers analyzed, all show the las system positively activating the rhl system, and none show the rhl system positively activating the las system.

Data Collection

We used three strains for the experimental observations: lasB:luxCDABE genomic reporter fusion in NPAO1∆lasI∆rhlI, lasI:luxCDABE genomic reporter fusion in NPAO1∆lasI∆rhlI, and rhlI:luxCDABE genomic reporter fusion in NPAO1∆lasI∆rhlI. The transcriptional reporters are based on the mini-CTX-lux reporters (Becher and Schweizer 2000). Promoter regions of the genes of interest were cloned upstream of the lux operon, creating a transcriptional fusion reporter. The CTX system ensures a single chromosomally integrated reporter. We streaked out all strains in Luria-Bertani (LB) agar at 37°C for 24 hours and then subcultured a single colony in 10 mL LB, incubated at 37°C under shaking conditions (180 rpm) for 24 hours.

We prepared 3‑oxo‑C₁₂‑HSL and C₄‑HSL in methanol at 7 different concentrations: 0.1, 0.5, 1, 2, 3, 4 and 5 µM, each diluted from 100 mM stock. We centrifuged all cultures and washed each three times using PBS. We then re-suspended in LB and diluted to an OD (600) of 0.05. We then transferred 200 µl of each culture to a black 96-well plate with a clear bottom and inoculated with signals at the indicated concentrations. We repeated each experiment to generate five replicates. Methanol with no signal was used as a control. The plates were incubated in BioSpa at 37°c for 18 h. Measurements of OD (600) and RLU (Relative Luminescence Units) were collected every hour.

Data Analysis

We estimated parameter values in Tables C, D, E, and F in S1 Text with non-linear regression by least squares using the Gauss-Newton algorithm (Ratkowsky 1983). Observations were limited to time ranges with peak expression values. (See Figs A, B, and C in S1 Text for detailed time course analysis.) Comparisons of model predictions and observed values are shown in Figs E, F, and G in S1 Text. Equilibrium values shown in Fig 5 were computed using a Trust-Region-Dogleg Algorithm (Powell 1968). Analyses performed and data visualizations created with MATLAB: Version 9.13 (R2022b) from The MathWorks Inc., Natick, MA and Stata Statistical Software: Release 17 from StataCorp LLC, College Station, TX. All original code is available on GitHub at https://github.com/GaTechBrownLab.

Additional third-party modules:

Jann B. “PALETTES: Stata module to provide color palettes, symbol palettes, and line pattern palettes,” Statistical Software Components S458444, Boston College Department of Economics, 2017, revised 27 May 2020.

Jann B. “COLRSPACE: Stata module providing a class-based color management system in Mata,” Statistical Software Components S458597, Boston College Department of Economics, 2019, revised 06 Jun 2020.

Jann B. “HEATPLOT: Stata module to create heat plots and hexagon plots,” Statistical Software Components S458598, Boston College Department of Economics, 2019, revised 13 Oct 2020.

Custom color schemes adapted from seaboarn.

Watson ML. “seaborn: statistical data visualization.” Journal of Open Source Software 2021 Apr 06;6(60): doi: 10.21105/joss.03021.

Acknowledgements: We are grateful to Diego Sunga and Prisha Sharma for their support in conducting follow-up experiments. We thank Steve Diggle, Marvin Whitely, Joshua Weitz and members of the Center for Microbial Dynamics and Infection (CMDI) for valuable comments and discussion on this work.

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