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Parameters, architecture and emergent properties of the Pseudomonas aeruginosa LasI/LasR quorum-sensing circuit

Martin Schuster

Martin Schuster

Department of Microbiology, Oregon State University, Corvallis, OR 97331, USA

[email protected]

Contribution: Conceptualization, Formal analysis, Methodology, Resources, Writing – original draft

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Christina Li

Christina Li

Department of Microbiology, Oregon State University, Corvallis, OR 97331, USA

Contribution: Formal analysis, Investigation

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Parker Smith

Parker Smith

Department of Microbiology, Oregon State University, Corvallis, OR 97331, USA

Contribution: Investigation, Methodology

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Christina Kuttler

Christina Kuttler

Department of Mathematics, Technische Universität München, 85748 Garching, Germany

Contribution: Formal analysis, Methodology, Resources, Writing – review & editing

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    Abstract

    Quorum sensing is a widespread process in bacteria that controls collective behaviours in response to cell density. Populations of cells coordinate gene expression through the perception of self-produced chemical signals. Although this process is well-characterized genetically and biochemically, quantitative information about network properties, including induction dynamics and steady-state behaviour, is scarce. Here we integrate experiments with mathematical modelling to quantitatively analyse the LasI/LasR quorum sensing pathway in the opportunistic pathogen Pseudomonas aeruginosa. We determine key kinetic parameters of the pathway and, using the parametrized model, show that quorum sensing behaves as a bistable hysteretic switch, with stable on and off states. We investigate the significance of feedback architecture and find that positive feedback on signal production is critical for induction dynamics and bistability, whereas positive feedback on receptor expression and negative feedback on signal production play a minor role. Taken together, our data-based modelling approach reveals fundamental and emergent properties of a bacterial quorum sensing circuit, and provides evidence that native quorum sensing can indeed function as the gene expression switch it is commonly perceived to be.

    1. Introduction

    Cell–cell communication, termed quorum sensing (QS), is a common process in bacteria that regulates collective behaviours according to population cell density by producing and sensing diffusible autoinducer (AI) molecules [1,2]. QS-controlled behaviours range from interactions with eukaryotic hosts and microbial warfare to protein secretion and biofilm formation. In Gram-negative bacteria, the basic QS circuitry consists of a LuxI-type synthase that produces an acyl-homoserine lactone (AHL) signal, and a cognate LuxR-type signal receptor [3,4]. The diffusible AHL signal accumulates during growth, binds to the LuxR-type receptor and activates the expression of target genes, as well as the expression of the synthase itself.

    QS has been extensively characterized genetically and biochemically, revealing important information about the function and regulation of individual components, target genes, induction kinetics, and signal specificity [46]. QS via AHL is particularly well understood in the Gram-negative model organism and opportunistic human pathogen Pseudomonas aeruginosa [5,6]. In P. aeruginosa, QS controls hundreds of genes, including extracellular enzymes, toxins, metabolites and biofilm components [79]. The major AHL system (termed ‘las’) consists of the LuxI-type synthase LasI that produces 3-oxo-dodecanoyl-homoserine lactone (3OC12-HSL) and the LuxR-type intracellular receptor LasR that binds the signal and dimerizes to activate transcription of target genes (figure 1). One of these target genes is lasI, creating a positive feedback loop [10]. For lasR, evidence of positive feedback-control is less clear [7,11,12]. In addition, the transcription of lasI is subject to negative regulation by the repressor RsaL, which is itself induced by LasR-3OC12-HSL, dampening signal accumulation during culture growth [13,14]. Movement of 3OC12-HSL across the cell envelope is probably by a combination of diffusion and active transport [15,16]. In this study, we will focus our analysis on the las QS system, acknowledging that there are other, subordinate QS systems (rhl, PQS) in P. aeruginosa [6].

    Figure 1.

    Figure 1. Pseudomonas aeruginosa las QS circuit. Red symbols indicate genes, blue symbols indicate proteins, and green symbols indicate small molecules. Se and Si denote extracellular and intracellular 3OC12-HSL signal, respectively; target represents one of many target genes in the LasR regulon. Shown are signal exchange, receptor dimerization upon signal binding, positive feedback on lasI and lasR, as well as negative feedback on lasI and rsaL. Black boxes indicate LasR binding sites, and the open box in the lasI-rsaL bidirectional promoter indicates an RsaL binding site.

    As described, the P. aeruginosa las system enables coordinated gene expression control at the population level, where the accumulation of extracellular 3OC12-HSL produced by all cells triggers a concerted response [17,18]. This behaviour is consistent with the general perception of QS as a population switch [1921], although some systems also exhibit heterogeneous responses with considerable cell-to-cell variation [2226].

    In addition to experimental inquiry, QS has been investigated from a theoretical perspective. Mechanistic mathematical modelling studies have revealed important insights into architecture and emergent properties of the signalling network [2736]. Theory predicts that the P. aeruginosa las QS system, with positive feedback and receptor dimerization as two key elements, may constitute a bistable system with two stable states, on and off [30]. Bistability has been considered a hallmark of the canonical QS circuitry, yet it has rarely been demonstrated experimentally [2729]. The only two cases we are aware of employ a non-native heterologous host system [26,37]. An important accompanying property of bistability is hysteresis, the dependency of the steady-state response on the direction of the parameter change. Hysteresis has functional implications by increasing the robustness of the QS response to fluctuations in environmental signal levels [26].

    In this study, we have combined experimental and theoretical approaches for a data-centred analysis of the P. aeruginosa las QS system. We collected extensive experimental data, built a mechanistic mathematical model, and fit the model to the data. Using the parametrized model, we investigated emergent properties including bistability and hysteresis, as well as the role of positive and negative feedback. Our findings help define the design principles and functional capacity of QS systems.

    2. Methods

    2.1. Bacterial strains, plasmids and growth conditions

    We used the P. aeruginosa PAO1 wild-type strain (WT, Iglewski lineage) [38], and an isogenic mutant containing a markerless in-frame deletion in lasI [18]. We used the promoter-probe vector pPROBE.AT [39] with lasI’-gfp, lasR’-gfp and rsaL’-gfp transcriptional fusions, respectively. The encoded green fluorescent protein (GFP) variant is the stable and fast-folding gfpmut1 [40]. The lasI’-gfp and rsaL’-gfp fusion constructs were described previously [18], whereas the lasR'-gfp fusion was constructed for the present study, using routine molecular cloning techniques. A 333 bp lasR promoter region was PCR-amplified from the PAO1 genome using forward primer 5′-N6AAGCTTGCGATGGGCCGACAGTGAAC-3′ and reverse primer 5′-N6GAATTCTCTTAAACTATTAACCAATCAGCCAAATATGGATT-3′. The PCR product was digested with HindIII and EcoRI (restriction sites underlined) and ligated with equally digested pProbe-AT. The resulting construct was confirmed by sequencing. Plasmids were introduced into chemically competent P. aeruginosa strains according to standard procedures [41]. Strains were grown at 37°C with shaking in Lennox Lysogeny Broth (LB) liquid cultures buffered with 50 mM 3-(N-morpholino)-propanesulfonic acid (MOPS), pH 7.0, or on LB agar plates. The antibiotic carbenicillin was added at a final concentration of 200 µg ml−1 for plasmid selection and maintenance. Growth was measured as optical density at 600 nm (OD600) in a photometer (Eppendorf Biophotometer), and as colony-forming units (CFU) per ml by dilution-plating. The conversion factor between OD600 and CFU ml−1 was determined by linear regression (electronic supplementary material, figure S1).

    2.2. Gene expression and AHL measurements

    We conducted gene expression and AHL quantitation of P. aeruginosa strains in LB-MOPS liquid cultures. We employed a pre-culturing scheme as described previously to dilute pre-existing GFP in cells to background levels [18]. In short, we inoculated pre-cultures with freshly streaked P. aeruginosa colonies, and grew them to an OD600 of 0.1 to 0.2. Using these pre-cultures, we inoculated experimental cultures to an initial OD600 of 0.001. Cultures were grown for 3 h before the first sampling. Subsequent samples were taken every 0.5 to 1 h, for up to 12h of total cultivation time. For gene expression analysis, P. aeruginosa WT and lasI mutant strains contained lasI’-gfp, lasR’-gfp or rsaL’-gfp reporter fusions. At each time point, aliquots were transferred to a black-walled microtiter plate to quantify GFP intensity in a fluorescence plate reader (Tecan M200) at 480 nm excitation and 535 nm emission wavelengths. The fluorescence of LB-MOPS medium was subtracted as a blank. The net fluorescence values were divided by the OD600 at the respective time point to attain relative fluorescence intensities.

    For AHL analysis, at each time point a 5 ml culture aliquot was extracted twice with acidified ethyl acetate as described in [42,43]. Extracts were concentrated to dryness by evaporation and resuspended in fresh ethyl acetate. The 3OC12-HSL content in these extracts was determined by an established Escherichia coli bioassay containing lasR and a lasB’-lacZ reporter [42,43]. ß-galactosidase activity of bioassay cultures was measured using the Galacto Light Plus Reporter Gene Assay (Thermo Fisher). 3OC12-HSL concentrations in the experimental samples were determined by comparison with a standard curve generated with synthetic 3OC12-HSL (RTI International).

    2.3. Mathematical model

    Mathematical modelling was based on a previously described quantitative model of QS [28,44]. This model consists of a system of ordinary differential equations (ODEs) that describes the change in signal and protein components, as well as cell growth, over time. Parameters are described in table 1. We considered the cellular signalling dynamics with two different versions of this model: (i) a minimal model with positive feedback on lasI expression only, and (ii) an extended model with positive feedback on both lasI and lasR expression, as well as negative feedback on lasI and rsaL expression via RsaL. The minimal model is described as follows:

    extracellular 3OC12-HSL, Se

    dSedt=ρvcsec(SiSe)γexSe,

    intracellular 3OC12-HSL, Si

    dSidt=csec(SeSi)+AsynI

    and LasI synthase, I

    dIdt=(VImaxVImin)SimSim+KSm+VIminγII.
    The minimal model is virtually identical to that by Fujimoto and Sawai [28]. It makes several assumptions and simplifications. It models the positive feedback of 3OC12-HSL on lasI expression directly through the Hill function Sim/(Sim+KSm) and does not explicitly model LasR. However, it considers positive cooperativity (expressed by the Hill coefficient m), which primarily manifests through receptor dimerization [49]. The binding of one signal molecule to a receptor monomer, which then drives dimerization, is assumed to be non-cooperative, as is the binding of LasR-3OC12-HSL to target promoters [53,54]. The model omits the concentrations of mRNA species, as they are assumed to be in quasi-steady state with the respective protein concentrations [27]. It considers extracellular but not intracellular signal degradation, because the contribution of intracellular degradation is negligible, given the extremely small intracellular volume fraction. The model also does not distinguish between contributions of diffusion and active transport to the movement of signal in and out of the cell [16], and does not explicitly consider potential effects of growth rate on the concentrations of intracellular species [55]. These simplifications reduce the complexity of the model such that parameter estimation is computationally feasible. The cellular volume fraction (ρv) affects the extracellular 3OC12-HSL concentration (Se) through population growth, as described below.

    Table 1. Model parameters.

    quantity name start valuea unit reference estimated valuec
    minimal model extended model
    fittable parameters
    VI max maximum LasI synthase expression rate 1.9 μmol (l h)−1 [31] 4.2 2.7
    VI min minimum LasI synthase expression rate 0.022 μmol (l h)−1 [31] 0.020 0.017
    VR max maximum LasR receptor expression rate 1 μmol (l h)−1 this study 1.3
    VR min minimum LasR receptor expression rate 0.1 μmol (l h)−1 this study 0.27
    VL max maximum RsaL repressor expression rate 1 μmol (l h)−1 this study 1.0
    VL min minimum RsaL repressor expression rate 0.1 μmol (l h)−1 this study 0.0025
    KS QS induction threshold 0.1 μmol l−1; μmol2 l−2 b [17,18] 0.13 0.11
    KL1 repression threshold for lasI 1 μmol l−1 this study 2.2
    KL2 repression threshold for rsaL 1 μmol l−1 this study 2.8
    γex extracellular signal degradation rate 0.0174 h−1 [45] 0.0029 0.0014
    csec AI transport rate across cell envelope 12 h−1 [16,28] 88 78
    Asyn synthesis rate of autoinducer 1.6 h−1 [46,47] 5.5 4.3
    γI degradation rate of synthase 0.36 h−1 [31] 1.6 0.78
    γR degradation rate of receptor 0.36 h−1 [31] 0.42
    γL degradation rate of repressor 0.36 h−1 [31] 0.093
    γG degradation rate of GFP 0.029 h−1 [48] 0.79 0.12
    μ growth rate constant 1.5 h−1 this study 1.3 1.2
    K carrying capacity 3 × 1012 CFU l−1 this study 2.3 × 1012 2.4 × 1012
    τ scaling factor for GFP expression 10 000 Rel. GFP intensity/(μmol l−1) this study 10 000 10 000
    m Hill coefficient for induction 2 none [28,49] 2.3 2.2
    n Hill coefficient for repression 2 none [50] 6.8
    other parameters
    Vcell volume of a single cell 6 × 10−16 l [51,52] n/a n/a
    Vtot total culture volume 1 l this study n/a n/a
    ρv volume fraction of cells assigned none none n/a n/a

    aStart values denote the initial parameter values used in the fitting attempt. In addition to these start values, we defined the initial conditions for the variables in the ODE system at time zero. These were 0.01 µmol l−1 for I, R, L and Si, 100 relative fluorescence intensity units for GI, GL, and GR, and 109 CFU l−1 for P.

    bUnits are μmol l−1 for the minimal model, and μmol2 l−2 for the extended model.

    cRounded to two significant figures.

    In addition to the minimal model, we built the following extended model that incorporates additional protein components and feedback regulation:

    extracellular 3OC12-HSL, Se

    dSedt=ρvcsec(SiSe)γexSe,

    intracellular 3OC12-HSL, Si

    dSidt=csec(SeSi)+AsynI,

    LasI synthase, I

    dIdt=(VImaxVImin)(RSi)m(RSi)m+KSmKL1nLn+KL1n+VIminγII,

    LasR receptor, R

    dRdt=(VRmaxVRmin)(RSi)m(RSi)m+KSm+VRminγRR
    and RsaL repressor, L
    dLdt=(VLmaxVLmin)(RSi)m(RSi)m+KSmKL2nLn+KL2n+VLminγLL.
    The full model includes LasR and RsaL, assuming separate basal and maximum synthesis rates (Vmin and Vmax) for each protein. Here, the Hill function models the dual positive feedback of signal–receptor complex formation on both signal synthase and receptor expression, following a previously proposed notation [28,44]. It considers positive cooperativity through receptor dimerization as above, and the same induction threshold (KS) for all three proteins, which is consistent with our experimental data (see below). The repression of lasI and rsaL by RsaL is modelled by the term KLn/(Ln+KLn), with separate repression constants (KL1 and KL2), respectively. This distinction in repression strength is again motivated by our experimental data and is consistent with the asymmetric promoter architecture (even though there is only a single RsaL binding site in the lasI-rsaL bidirectional promoter, it is positioned such that disproportionate repression is conceivable) [13]. A Hill coefficient (n) allows for cooperativity in the form of RsaL dimerization [50].

    Further, the ODE for extracellular 3OC12-HSL is simplified in both models by assuming a quasi-steady state between intracellular and extracellular signal. Hence, Se can be expressed through Si

    dSedt=ρvcsec(SiSe)γexSe=0Se=cseccsec+γex/ρvSi.

    Each cellular model is connected to a growing population as follows.

    population density, P

    dPdt=μP(KP)K
    and volume fraction, ρv
    ρv=NVcellVtot=PVcell.
    Population density P is modelled with a logistic growth equation. P is defined as cell number N (in colony-forming units, CFU) per total culture volume Vtot (normalized to 1 l). The volume fraction ρv links population growth to the single-cell model through a change in Se. Strictly speaking, the volume fraction ρv is a volume ratio N(Vcell)/(VtotVcell), which can be readily shown by mass conservation law. However, because Vcell is very small compared with Vtot, VtotVtotVcell.

    Finally, we explicitly model GFP intensities to relate our measurements of lasI’-gfp, lasR’-gfp and rsaL’-gfp reporter expression to the actual protein components in the cell (only lasI’-gfp in the reduced model, and all three reporters in the full model).

    GFPlasI,GIdGIdt=τ((VImaxVImin)(RSi)m(RSi)m+KSmKL1nLn+KL1n+VImin)γGGI,
    GFPlasR,GRdGRdt=τ((VRmaxVRmin)(RSi)m(RSi)m+KSm+VRmin)γGGR
    and GFPrsaL, GL
    dGLdt=τ((VLmaxVLmin)(RSi)m(RSi)m+KSmKL2nLn+KL2n+VLmin)γGGL.
    The factor τ relates the rate of GFP synthesis to that of the respective protein. We assume here that promoter activity is proportional to protein concentration. We believe this assumption is appropriate given that QS is a transcriptional regulatory pathway, but of course it neglects possible post-transcriptional effects. The apparent degradation rate of GFP (γG) is the same in all equations.

    2.4. Model analysis and parameter estimation

    The model was built, simulated and analysed with the complex pathway simulator software package (COPASI 4.29) [56,57]. Parameters were estimated with the parameter estimation function (method Levenberg-Marquardt), using the experimental data in figure 1 and the guessed initial parameter values shown in table 1. Parameter fitting was done in iterations, starting with the initial parameter set and then again with the parameter set obtained after the initial step. The fitting algorithm calculates the following objective function, to which each experiment contributes with a weighted sum of squares [57]:

    E(P)=i.jωj(xi,jyi,j(P))2,
    where P is the tested parameter set, xi.j is a point in the dataset, and yi.j is the corresponding simulated value. The indices i and j denote rows and columns in the dataset, corresponding to the measured time points and quantities, respectively. The weight for each data column is given by ωj. Weights were calculated using the mean squares method.

    The Akaike information criterion (AIC) as an estimator of the prediction error was calculated as

    AIC=n(lnSSEn)+2k,
    where n is the number of data points in the dataset, k is the number of parameters fitted plus one and SSE is the least-squares error, i.e. the sum of the objective function in table 2 [58].

    Table 2. Parameter estimation statistics.

    fitted value data points minimal model
    extended model
    objective function RMSEa objective function RMSEa
    CFU l−1 15 0.024 0.040 0.014 0.031
    [3OC12-HSL] 8 0.80 0.32 0.053 0.082
    GFPlasI 15 0.080 0.073 0.034 0.048
    GFPlasR 15 0.092 0.078
    GFPrsaL 17 0.038 0.048
    sum 70 0.91 0.23

    aRMSE, root mean square error.

    A steady-state analysis was conducted with the parameter scan function in Copasi. The GFPlasI level at steady state was determined with cell density as the control variable. To maintain a constant cell density, the growth rate was set to zero, and time courses were run for 1000 h at 1000 different cell density values within a given range. Each cell density value was held constant throughout each time course. Two different initial LasI and GFPlasI concentrations were set to separately investigate the off-to-on transition (I0 = 0.01, GI 0 = 100) and the on-to-off transition (I0 = 1, GI0 = 10 000).

    2.5. Feedback analysis

    To investigate the significance of feedback in the model, we employed the parametrized extended QS model. In addition to Si, we evaluated effects on the actual circuitry components I, R and L directly, rather than on GI, GR and GL. To essentially eliminate each one of the three feedback loops individually, we set the equilibrium constants in the respective Hill and repression functions to very large values (1 × 106), namely: Ks in the equation for I to eliminate positive feedback on I, Ks in the equation for R to eliminate positive feedback on R, and KL1 and KL2 in the equations for I and L to eliminate negative feedback on I and L via RsaL. Further, we added a hypothetical QS-controlled protein T to the model to evaluate induction kinetics and steady-state behaviour independent of individual feedback manipulations that alter the kinetics of I, R and L. It follows I induction kinetics but does not include negative influence from RsaL, and the value for Ks remains at 0.11 µmol2 l−2 (table 1).

    dTdt=(VImaxVImin)(RSi)m(RSi)m+KSm+VIminγIT.

    3. Results

    3.1. Measurement of gene expression and signal production

    As a first step, we collected the experimental data required to describe the las QS circuit. We focused on species relevant to our mathematical model, and on experimental methods that are feasible and quantitative. We measured the expression of lasI, lasR and rsaL in P. aeruginosa via GFP reporter fusions, and we measured the concentration of extracellular 3OC12-HSL by bioassay (figure 2). We used a standard growth medium, buffered Lysogeny broth, that is most commonly used in the field and allows comparison with other studies. We collected time courses across the full range of QS induction dynamics, from basal expression in the off state to maximal expression in the on state.

    Figure 2.

    Figure 2. Cell density, signal production, and gene expression in the P. aeruginosa las QS circuit. P. aeruginosa strains were grown in LB-MOPS batch cultures. Cell density (OD600) and 3OC12-HSL production were measured in the WT strain. Gene expression was measured as GFP fluorescence intensity in WT and lasI mutant strains containing lasI’-gfp, rsaL’-gfp or lasR’-gfp reporter constructs, respectively. GFP intensity was normalized to cell density. Error bars indicate s.d. (n = 3).

    Consistent with previous studies, we observed a sharp cell density-dependent increase in GFPlasI and GFPrsaL from initially very low, basal levels in the WT strain. The expression of GFPlasR increased concurrently, although basal expression levels were higher. For comparison, the expression of all GFP reporter fusions was much lower in the lasI deletion mutant, confirming that their induction is QS-dependent. By contrast to GFPlasI and GFPrsaL, GFPlasR showed a modest, partially QS-independent increase throughout growth, the implications of which are discussed below. As expected, the observed concentrations of 3OC12-HSL during culture growth of the WT correlated well with the expression levels of GFPlasI.

    3.2. Mathematical modelling and parameter estimation

    Next, we sought to fit our experimental data to a mechanistic mathematical model of QS. To establish a baseline, we first considered a minimal QS model with single feedback, introduced by Fujimoto & Sawai [28]. This ODE model includes four species, cell density, LasI concentration, extracellular 3OC12-HSL concentration and intracellular 3OC12-HSL concentration. It models positive feedback on lasI directly via changes in signal concentration (figure 3 top left; see Methods), with an underlying quasi-steady-state assumption. We amended the model with a GFP equation that directly relates gene expression measured by lasI’-gfp fusion to the concentration of LasI in the cell. Hence, three model variables, cell density, extracellular 3OC12-HSL and GFPlasI, could be directly fit to experimental data. We asked whether this model is sufficiently complex to describe the experimentally observed induction dynamics. We used the modelling software Copasi [56] to implement the model and estimate parameters using a least-squares method. To guide parameter estimation, start values were taken from our own experimental data when available, or taken from the literature (table 1). In addition to the cell growth data, the minimal model was able to fit the timing of induction well, but failed to accurately capture the subsequent decrease in both 3OC12-HSL and GFPlasI levels (figure 3 and table 2). We reasoned that the negative feedback on lasI by RsaL could largely account for the discrepancy.

    Figure 3.

    Figure 3. Parametrized models fit to experimental data. The minimal model is shown on the left, and the extended model is shown on the right. Black symbols, experimental data (means from figure 1 with n = 3); red line, model fit.

    As a second step, we conducted a parameter estimation with the extended QS model, which incorporates repression of both lasI and rsaL by RsaL (figure 3 top right). The repression of both genes works by the binding of an RsaL dimer to a conserved sequence element in the lasI-rsaL bidirectional promoter [13,14,50]. The extended model also incorporates LasR and its regulation by a second, positive feedback loop, which is supported by our experimental data. We performed a parameter estimation with the extended model as described above and obtained a greatly improved fit that captured the decline in 3OC12-HSL and GFPlasI levels late in growth. Compared with the minimal model, the extended model showed decreased RMSE values for 3OC12-HSL and GFPlasI (table 2). The RMSE for the newly fit GFPrsaL was equally low, while that for GFPlasR was somewhat higher, possibly indicating additional layers of regulation not captured with our model (see Discussion). Many estimated parameters were remarkably close to the start values used to seed the model (table 1).

    To estimate the relative quality of each model, we determined the Akaike information criterion (AIC), which weighs the goodness of fit against the number of parameters used to achieve that fit. The AIC is −116 for the minimal model, and −181 for the extended model (electronic supplementary material, table S1). Because a smaller number indicates higher model quality, we conclude that the extended model performs better, even when considering the larger parameter space.

    3.3. Bistability and hysteresis

    Having parametrized the extended model, we sought to investigate two important and related emergent properties of the QS network. We asked whether the extended model, as fit to experimental data, can function as a bistable switch with two stable states (on and off), and whether this switch exhibits hysteresis. To answer these questions, we modelled signal synthase expression, GFPlasI, at cellular steady state, as a function of different cell densities (figure 4). In support of bistability and hysteresis, we observed two distinct on and off states without a graded transition in between, and we observed sharp transitions at different cell densities, depending on the direction of the change. The cell density range between the low → high and the high → low transition is sizeable, suggesting that hysteresis could significantly improve the stability of the QS response in a fluctuating environment.

    Figure 4.

    Figure 4. Bistability and hysteresis. Steady-state GFPlasI expression at varying cell densities according to the extended model. This plot represents stable equilibrium conditions, with dGI/dt=0 and dP/dt=0, respectively. Bistability is apparent as two distinct expression states (low or high), lacking a graded transition. Hysteresis is apparent as distinct cell density thresholds (arrows) in the low-to-high and the high-to-low transition.

    Of note, the state switching modelled here is so-called group-level bistability triggered by the accumulation of extracellular signal in a growing population of cells [28]. A concerted group-level response is supported by recent single-cell gene expression data showing unimodal distribution of GFPlasI under culture conditions identical to those used here [17,18].

    3.4. The role of positive and negative feedback

    Next, we employed the parametrized extended model to analyse the significance of feedback architecture on the QS response. There are two positive feedback loops (LasR-3OC12-HSL activating lasI and lasR, respectively), and a negative feedback loop (RsaL activated by LasR-3OC12-HSL repressing lasI and rsaL). We removed one of the respective feedback loops in silico, as explained in Methods, and we simulated time courses for the concentrations of the actual QS circuitry components Se, I, R and L in the modified models (figure 5). To also evaluate these effects on a QS-controlled component that is not itself involved in positive or negative feedback, we added a separate, hypothetical target protein T to the model, and we simulated its behaviour under dynamic and steady-state conditions. In the P. aeruginosa LasR regulon, such a target protein candidate would be, for example, the Type VI secretion effector PAAR4, the alkaline protease AprA, or the nucleoside hydrolase Nuh [7,14,59,60].

    Figure 5.

    Figure 5. Feedback analysis. The time courses of all QS components, I (LasI), R (LasR), L (RsaL), Se (extracellular 3OC12-HSL), and a hypothetical target T, are shown on the left, and the steady-state behaviour of T is shown on the right. The parametrized extended model was simulated with and without positive or negative feedback loops, as indicated by the vertical text on the left. A break was introduced into the y-axis of the bottom left panel in order to keep the scale at the lower concentrations identical to that of the other panels.

    The omission of positive feedback on I had a large effect. Compared with WT conditions, it delayed the timing of induction, greatly decreased the concentrations of I and Se and more modestly those of the other components, and resulted in rheostat-like rather than bistable behaviour. Of note, basal expression of I still resulted in the gradual accumulation of Se to concentrations near the Ks (0.1 µM), such that R, L and T were induced to substantial levels.

    By contrast, the omission of positive feedback on R had a much smaller effect. It did not change the magnitude or timing of induction (except for R, of course) and still produced bistable behaviour, albeit with a narrower margin for hysteresis. The observation that positive feedback on I is essential for bistability is consistent with previous theoretical work [30], as is the observation that positive feedback on R enhances hysteresis [26].

    The omission of negative feedback on I and L via L had the expected effect on the magnitude of I, L and Se, but it did not affect the timing of induction, nor bistability in general. The increase in the levels of I, L and Se is in agreement with experimental data on lasI, rsaL and 3OC12-HSL expression in a P. aeruginosa rsaL mutant [14].

    4. Discussion

    This study brings to bear experimental and mathematical approaches for an integrative understanding of bacterial QS. Experimental and mathematical inquiry has largely co-existed in this field, arguably without much cross-fertilization. Consequently, a systems-level analysis of the functions of native AHL-QS systems in a variety of experimental model species such as Vibrio fischeri, P. aeruginosa, and Agrobacterium tumefaciens remains scarce, and important functional assumptions remain unproven. In this study, we mechanistically modelled the native LasI-LasR (las) QS system in P. aeruginosa. We fit the model to experimental time series data, and used the parametrized model to investigate feedback architecture and steady-state behaviour, linking network design to function. We found that the parametrized model constitutes a bistable switch with distinct on and off states, as well as hysteretic memory. We also found that positive feedback on signal production is critical in maintaining bistability.

    Our chosen modelling approach is one of many [35], but it was guided by the objective to mechanistically describe QS, and to reduce parameter space as much as possible while retaining the ability to explicitly model all experimentally measured variables. Indeed, we found that our extended model is sufficiently complex to accurately describe the experimentally observed QS dynamics. This is remarkable given that the QS circuitry in P. aeruginosa is embedded in global cellular regulation [5,6,61]. However, our focus was exclusively on the las system, and there is evidence that this system is more autonomous than the two other, interconnected QS systems, termed rhl and pqs. The las system is considered to be atop the QS hierarchy, and it is influenced less by environmental cues than the rhl system [62]. This does not mean that there are no other factors that influence the las system [5]. For example, the partially QS-independent increase in lasR expression observed experimentally (figure 2) [7,12] is not fully captured by the model. Implementing a clear mechanistic basis for this pattern is not straightforward, as the underlying pathway and level of regulation under the specific culture conditions is not clear. However, it is evident from our feedback analysis that positive feedback on LasR itself is not a critical component of the QS circuitry in P. aeruginosa, such that the molecular details of LasR expression are of lesser importance.

    Instead, we find that positive signal feedback on lasI is required for bistability. This result is consistent with theoretical work by Goryachev et al. [29,30]. Goryachev et al. also reasoned that a second network element in addition to the classical positive signal feedback is necessary for bistable behaviour. In our network, this second element probably is LasR receptor dimerization. Receptor multimerization is one molecular interaction that contributes to positive cooperativity with a Hill coefficient greater than 1 as estimated in our model. By contrast, experimental and theoretical work by Haseltine and Arnold found that a synthetic QS circuitry with V. fischeri components in E. coli only exhibits bistability if both I and R are under positive feedback control [37]. Accordingly, they estimated the Hill coefficient to be 1 in their model (no cooperativity), which neglects V. fischeri LuxR receptor dimerization [63], but is consistent with the experimentally determined binding characteristics of LuxR to target DNA [64].

    Finally, the omission of negative feedback via RsaL had no major effect on induction dynamics and bistable behaviour. This result indicates that RsaL-mediated repression could mainly function to dampen excessively high signal production, possibly as a cost-saving mechanism, although it is known that RsaL represses many other genes in P. aeruginosa besides lasI and rsaL [14].

    It will be interesting to test some of the modelling predictions made here with specific gene deletions, promoter mutations, or target gene-reporter fusions, under dynamic or steady-state growth conditions. It will also be of interest to apply a similar data-based modelling approach to other native QS systems. Individual parameters will differ, and emergent properties may as well, given that a combination of factors (kinetic constants, positive and negative feedback, receptor dimerization and molecular noise) can all affect network behaviour [26,30]. In the case of individual target genes as well as complete QS systems, potential regulatory complexities will need to be considered.

    Taken together, our work estimates parameter values and rate constants, and defines design principles and emergent properties of a native QS system. This knowledge can help us quantify the performance of QS in various contexts: in bacterial communities and in association with a host, in the design of synthetic QS circuits with desired functions, and in the development of new therapeutic (quorum-quenching) drugs that target QS in bacterial pathogens.

    Data accessibility

    Additional experimental and modelling data are available in the electronic supplementary material [65].

    Authors' contributions

    M.S.: conceptualization, formal analysis, methodology, resources, writing—original draft; C.L.: formal analysis, investigation; P.S.: investigation, methodology; C.K.: formal analysis, methodology, resources, writing—review and editing.

    All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

    Conflict of interest declaration

    We declare we have no competing interests.

    Funding

    This work was supported by National Science Foundation grant MCB 2106212, by the Alexander von Humboldt Foundation, by the German Academic Exchange Service (all to M.S.), and by the Sheila van Zandt student fellowship (to C.L. and P.S.).

    Acknowledgements

    We thank Johannes Müller, Department of Mathematics, Technische Universität München, for initial simulations, discussions, and comments on the manuscript.

    Footnotes

    Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.6451363.

    Published by the Royal Society. All rights reserved.

    References