Title: Robust and sensitive control of a quorum‐sensing circuit by two interlocked feedback loops
Abstract: Article16 December 2008Open Access Robust and sensitive control of a quorum-sensing circuit by two interlocked feedback loops Joshua W Williams Joshua W Williams Department of Biological Sciences, Virginia Tech, Blacksburg, VA, USA Search for more papers by this author Xiaohui Cui Xiaohui Cui Department of Biological Sciences, Virginia Tech, Blacksburg, VA, USA Search for more papers by this author Andre Levchenko Corresponding Author Andre Levchenko Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD, USA Search for more papers by this author Ann M Stevens Corresponding Author Ann M Stevens Department of Biological Sciences, Virginia Tech, Blacksburg, VA, USA Search for more papers by this author Joshua W Williams Joshua W Williams Department of Biological Sciences, Virginia Tech, Blacksburg, VA, USA Search for more papers by this author Xiaohui Cui Xiaohui Cui Department of Biological Sciences, Virginia Tech, Blacksburg, VA, USA Search for more papers by this author Andre Levchenko Corresponding Author Andre Levchenko Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD, USA Search for more papers by this author Ann M Stevens Corresponding Author Ann M Stevens Department of Biological Sciences, Virginia Tech, Blacksburg, VA, USA Search for more papers by this author Author Information Joshua W Williams1, Xiaohui Cui1, Andre Levchenko 2 and Ann M Stevens 1 1Department of Biological Sciences, Virginia Tech, Blacksburg, VA, USA 2Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD, USA *Corresponding authors. Department of Biomedical Engineering, Johns Hopkins University, 208C Clark Hall, 3400 N Charles Street, Baltimore, MD 21218, USA. Tel.: +1 410 516 5584; Fax: +1 410 516 4771; E-mail: [email protected] of Biological Sciences, Virginia Tech, 219 Life Sciences 1 (0910), 2119 Derring Hall, Washington Street, Blacksburg, VA 24061, USA. Tel.: +1 540 231 9378; Fax: +1 540 231 4043; E-mail: [email protected] Molecular Systems Biology (2008)4:234https://doi.org/10.1038/msb.2008.70 PDFDownload PDF of article text and main figures. ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InMendeleyWechatReddit Figures & Info The quorum-sensing (QS) response of Vibrio fischeri involves a rapid switch between low and high induction states of the lux operon over a narrow concentration range of the autoinducer (AI) 3-oxo-hexanoyl-L-homoserine lactone. In this system, LuxR is an AI-dependent positive regulator of the lux operon, which encodes the AI synthase. This creates a positive feedback loop common in many bacterial species that exhibit QS-controlled gene expression. Applying a combination of modeling and experimental analyses, we provide evidence for a LuxR autoregulatory feedback loop that allows LuxR to increase its concentration in the cell during the switch to full lux activation. Using synthetic lux gene fragments, with or without the AI synthase gene, we show that the buildup of LuxR provides more sensitivity to increasing AI, and promotes the induction process. Elevated LuxR levels buffer against spurious variations in AI levels ensuring a robust response that endows the system with enhanced hysteresis. LuxR autoregulation also allows for two distinct responses within the same cell population. Synopsis The quorum-sensing (QS) circuitry of Vibrio fischeri is a model system for many other bacterial species that exhibit cell density-dependent gene regulation (Fuqua et al, 2001; Taga and Bassler, 2003; Waters and Bassler, 2005). In this system, LuxR functions as the receptor for the freely diffusible signal molecule 3-oxo-hexanoyl-L-homoserine lactone (AI), which is produced by LuxI (Choi and Greenberg, 1992; Lupp et al, 2003). Much is known about how this system becomes induced as cell density increases, activating a positive feedback loop on LuxI expression, thereby increasing signal production (Shadel and Baldwin, 1991). However, less is known about how the system maintains induction under conditions that would push the system toward the uninduced state. It has been hypothesized that this QS system should exhibit bistability and hysteresis in luminescence expression. In this study, we present both experimental and computational analyses demonstrating this behavior, as well as evidence for its underlying causes. Computational modeling of the luxR/I circuit indicated that a second positive feedback loop must be present for hysteresis in the QS response. The presence of the LuxI feedback loop alone is not sufficient to produce a response that is robust in the face of signal concentration changes, due to the diffusible nature of the AI. Because of this, we hypothesized that a LuxR-positive autoregulatory feedback loop, suggested in some earlier studies (Shadel and Baldwin, 1992), was indeed present and, in fact, was the second positive feedback loop in the system. To demonstrate the presence of this LuxR autoregulatory feedback loop, a recombinant strain of Escherichia coli expressing a circuit composed of luxR, with gfp in place of luxI (called lux01) was created, to precisely control the level of AI available for QS induction. By fully inducing this circuit, and removing the AI gradually through 25% hourly dilutions, as well as instantaneously through 25% serial dilutions, it was demonstrated that this circuit containing only LuxR was capable of a hysteretic dependence on AI (Figure 1). It was demonstrated, through qRT–PCR analysis of the quantity of luxR transcript, that this positive feedback was a result of AI-dependent LuxR transcriptional autoregulation. As cell density increases, and more AI accumulates in the cellular environment, LuxR–AI complex can form and activate transcription of both luxI and luxR. This allows the cell to increase its internal concentration of LuxR, and thereby maintain a higher concentration of LuxR–AI complex, which in effect allows for a higher internal concentration of the otherwise diffusible AI. Once the system becomes fully induced, this higher concentration of LuxR provides robustness to the system, and allows for maintenance of a high state of induction during fluctuations in AI concentration. This demonstrates that the level of LuxR in the cell is critical for maintaining a robust QS response, and that regulation on LuxR will play a major role in determining the state of the response. A key factor that is known to transcriptionally regulate LuxR is the cAMP–CRP complex, the active concentration of which depends on the available levels of PTS sugars such as glucose. In an environment containing higher concentrations of glucose, cAMP–CRP levels will be lower, and thus transcriptional activation of luxR will also decrease. Again, a combination of computational and experimental analyses was employed to analyze the effects of cAMP–CRP levels on the range of hysteresis that the lux01 circuit could achieve. As predicted, an increase in glucose in the medium caused a decrease in the maximum output of the response, as well as a decrease in the amount of bistability the system could exhibit. Importantly, as the levels of glucose increase, and thus, the level of LuxR decreases, more AI is required to enter into the bistable range of expression, but the upper limit of the bistable range (50 nM AI) is unchanged (Figure 3A–D). This again demonstrates that LuxR concentration and positive autoregulation are critical to the determination of the state of induction of the system. To gain a more detailed understanding of how induction varied across the population of cells, flow cytometry was used to measure single cell fluorescence during induction and AI dilution experiments. This analysis provided direct evidence for a bimodal distribution of the population, and also demonstrated that there are two stable states the system can achieve based on their history of AI exposure (Figure 2D). Through a combination of computational modeling and experimental evidence, we have shown hysteresis and bistability in the QS circuitry of V. fischeri. This behavior endows the system with robustness in luminescence expression when undergoing fluctuation in AI concentration, and allows for precise control of the QS response. This behavior is important to consider due to the fact that many pathogenic bacterial species control virulence factor expression through QS circuits homologous to the LuxR/I system, and this type of genetic expression system is widely used in synthetic biology. Introduction Quorum sensing (QS) is an example of cell–cell communication in bacteria, allowing an assemblage of closely positioned cells to alter its behavior in a coordinated manner, if the cell density exceeds a specific threshold. QS regulates a plethora of critically important phenotypes, including antibiotic production, release of exoenzymes, production of virulence factors, induction of genetic competency, conjugative plasmid transfer, biofilm formation and bioluminescence (Fuqua et al, 2001; Waters and Bassler, 2005; Reading and Sperandio, 2006). In addition to understanding the role of these bacterial phenotypes to pathogenic and symbiotic states, analysis of the mechanisms underlying QS might shed light on how the behavior of a single cell can be tightly and robustly coordinated with the behavior of the cell group. The QS response of Vibrio fischeri is a model system for many other QS systems that share networks similar to the LuxR/I network (Taga and Bassler, 2003). LuxR is an autoinducer (AI)-dependent positive regulator of the lux operon, and LuxI produces the AI molecule, 3-oxo-hexanoyl-L--homoserine lactone. Much is known about how the LuxR/I system achieves activation of the lux operon leading to bioluminescence. A number of factors, including the activator complex cAMP–CRP, regulate the expression of luxR (Friedrich and Greenberg, 1983; Dunlap and Greenberg, 1985, 1988). LuxR then activates expression of the lux operon when the concentration of LuxR–AI complexes reaches a critical threshold. This leads to higher levels of AI, generating a positive feedback loop (Dunlap and Greenberg, 1988; Choi and Greenberg, 1992; Stevens and Greenberg, 1999; Lupp et al, 2003). It has been proposed that LuxR not only regulates the lux operon but it might also positively or negatively autoregulate the QS response through modulating its own expression (Dunlap and Ray, 1989; Shadel and Baldwin, 1991, 1992; Chatterjee et al, 1996), although the precise molecular basis for this autoregulation remains unknown. The presence of one or more feedback interactions in the molecular networks underlying QS in V. fischeri and other bacterial systems might lead to such emergent properties as hysteretic responses and the associated ‘memory’ of the previous network states. Such memory-like properties have been suggested for other systems containing positive feedback interactions, based both on mathematical modeling and experimental investigation (Ferrell, 2002; Levchenko, 2003; Sha et al, 2003; ; Angeli et al, 2004; Ninfa and Mayo, 2004; Ozbudak et al, 2004). Some mathematical models of V. fischeri QS signaling have suggested the existence of hysteresis in response to extracellular AI concentration ([AI]), based on AI-induced luxI expression leading to a further increase in AI production (James et al, 2000; Cox et al, 2003; Goryachev et al, 2006; Muller et al, 2006). This positive feedback is reliant on retention and accumulation of AI in the cell milieu. However, as the AI can freely diffuse across the cell membrane (Kaplan and Greenberg, 1985) this can make the system vulnerable to potentially rapid changes in [AI]. In a classical hysteresis analysis, the [AI] is exogenously fixed at various levels. This can mask the feedback-based increase of [AI], and hence complicate the experimental and theoretical analysis of the effects of the positive LuxI-based feedback. Because the mechanism of LuxR autoregulation in V. fischeri is unknown, the analysis of the onset of QS has been based on various assumptions (Shadel and Baldwin, 1991, 1992; Minogue et al, 2002). If this autoregulation is indeed present, its role in the QS response is presently not clear. In this report, through a combination of modeling and experiments, we provide further evidence for the existence of this second feedback circuit in the luxR/I QS network. Moreover, we demonstrate the presence of hysteresis in a reduced lux network lacking LuxI-based feedback. In the network containing both feedback interactions, luxR-positive autoregulation can enhance response diversification and endow it with higher robustness to AI perturbation, thereby increasing the fidelity of the QS switch. Results Decoupling LuxI-mediated positive feedback A convenient way to computationally and experimentally analyze feedback loops is to decouple them. For the example of lux operon regulation, the [AI] can be held at a fixed value, and the resulting expression of the system can be determined. This analysis can be repeated for different [AI] values, yielding curves showing dependencies of luxI expression on AI ([LuxI]=f([AI])) or AI production on LuxI ([AI]=g([LuxI])). These curves, known as null clines of the underlying dynamical system, can intersect when plotted in the same coordinate system (e.g. by co-plotting f and g−1), revealing points corresponding to the steady-state concentrations of AI and LuxI in the reconstituted feedback system (Figure 1A). Depending on how nonlinear the functions f and g−1 are, they can intersect in various ways, yielding a different number of steady states, which can be either stable, (i.e. resistant to small concentration changes due to molecular noise or other random perturbations) or unstable (Angeli et al, 2004). Multiple steady states signal that the response can stably display different values, depending on the initial conditions of the systems (e.g. whether the initial [AI] is high or low). In the presence of many AI-secreting cells, the [AI] experienced by individual cells would be the sum of endogenously produced LuxI-mediated AI (AIin) and AI diffusing from other cells (AIex). Additionally, AIex can be controlled experimentally, by supplying synthetic AI. The presence of AIex can further lead to a shift of the g null cline by uniform addition of AIex: g′=g+[AIex]. This shift can lead to a change in the number of the steady states, allowing for a fast transition from low to high values of the response. For the complete lux operon, this would imply the frequently assumed hysteretic (history dependent) behavior, and a rapid onset of QS beyond a critical cell density. Figure 1.Analysis of GFP expression in the lux01 circuit in response to different AI concentrations. (A) An illustrative cartoon of a decoupled LuxI–AI-positive feedback system. One can find possible stable steady-state responses in a positive feedback circuit by separately analyzing how AI regulates LuxI and how LuxI regulates AI. The resulting dependencies, or null clines: f([AI]) and g([LuxI]), can be plotted together, with the intersection points yielding the steady states, which can be stable (closed circles) or not (open circle). (B) Analysis of the f([AI]) null cline by allowing the response of the lux01 circuit to reach steady states after the addition of different exogenously added [AI], for different amounts of time. The data are expressed as relative fluorescence units/optical density of the culture. Error bars represent the standard deviation between two independent triplicate data sets. (C) Analysis of the f([AI]) null cline by allowing the response of the lux01 circuit to reach steady state after dilution from high initial induction values (reached at 8 h after instantaneous 25% serial dilution from 6 h incubation in 100 nM [AI]) (blue triangles). This response is overlaid with the results of the 6 h incubation obtained in (B) (closed circles) and the area between the curves is shaded dark gray. The area between the 2 and 6 h induction curves in (C) is shaded light gray. The results of the 25% hourly dilution experiments from cultures initially induced for 2 h at different exogenous [AI] (30 nM, inverted triangles; 50 nM, squares; 100 nM, rhombi) prior to dilution are also shown. Error bars represent the standard deviation between two independent triplicate data sets. Download figure Download PowerPoint Previous analyses suggested that at least one of the null clines f or g−1 should be sufficiently nonlinear for multiple steady states to occur (Ferrell, 2002; Tyson et al, 2003; Angeli et al, 2004). To determine the properties of the f null cline, an Escherichia coli strain was created with a chromosomal insertion of a synthetic lux01 genetic circuit capable of expressing LuxR, but with a truncated divergently transcribed lux operon, so that all of the transcripts normally downstream of the promoter are replaced with gfp. Thus exogenous AI is required for GFP expression (Supplementary Figure S1). This allowed decoupling of the AI-LuxI feedback and experimental measurement of the f null cline under different conditions. Using the lux01 circuit, the steady-state GFP expression values for various [AI] were determined by increasing AI from 0 up to 100 nM (Figure 1B). The resulting nonlinear steady-state response curves provided an estimate of the true f null cline (henceforth referred to as fL). This experiment also established that the lux01 circuit became maximally induced after about 6 h under the conditions used. When AI was diluted from the media of the cells in a highly induced state (7 h incubation at 100 nM AI), either instantaneously to lower [AI] or through hourly 25% dilutions, the resulting, virtually identical response curves were distinct from fL in the 0–50 nM AI range when compared with cultures that were induced but not diluted (this higher response curve, referred to as fH, was identical with fL in the 50–100 nM AI range; Figure 1C). Furthermore, when AIex was gradually diluted at an hourly rate of 25% from cells stimulated to less than the maximally induced state (2 h incubation at 30, 50 or 100 nM), cellular GFP concentration transiently increased over time in all three cases, converging to and closely following the upper response curve fH (Figure 1C). This suggests that the cells remained in the higher state of induction while undergoing continual loss of AIex, until a critical threshold was met, at which point the cells began to return to the lower induction state. In combination, these results suggested that the response could stably converge to different response curves, fL and fH, following different types of lux01circuit induction, and thus that the particular shape of the f null cline was dependent on the initial induction state of the cells. LuxR-mediated positive feedback is supported by experimentation and modeling It has been proposed earlier that LuxR can transcriptionally regulate its own expression (Dunlap and Ray, 1989; Shadel and Baldwin, 1991, 1992; Chatterjee et al, 1996). To determine the dependence of luxR transcription on [AIex], qRT–PCR analysis was performed on RNA samples extracted from cells with steady-state GFP expression corresponding to the lower branch of the hysteresis graph (i.e. induced from 0 to maximum with 100 nM AI; Figure 1C). In the range of 0–30 nM AI, luxR transcription varied as a function of exogenous AI, and there was an increase in luxR mRNA that reached a maximum level at exogenous [AI] above 10 nM (Figure 2). These findings are consistent with the proposed AI-dependent LuxR-controlled hysteretic autoregulation of luxR and support the hypothesis that the second feedback loop suggested by the data could be due to positive LuxR transcriptional autoregulation. Figure 2.Validation of induced luxR expression by qRT–PCR analysis of luxR transcript levels in the lux01 circuit. qRT–PCR analysis of luxR transcript quantity was performed on undiluted MG1655-01S induced for 6 h at [AI] of 5, 10, 30, 50 and 100 nM. Error bars represent the standard deviation between three independent triplicate data sets. Asterisks indicate data values that are significantly higher than that of the 5 nM data with a P<0.025 (standard t-test). Download figure Download PowerPoint To further bolster this hypothesis and explore the underlying putative regulatory mechanism, a simple mathematical model of LuxR autoregulation in the lux01 circuit was developed. This model investigated two distinct possibilities: the presence and absence of LuxR-positive autoregulation encoded in this simplified lux network using two corresponding systems of ordinary differential equations. In both cases, plus or minus LuxR-positive autoregulation, the corresponding models could be treated analytically, allowing us to derive a number of conclusions without explicit definition of the model parameter values, many of which are still unknown. The model postulating a positive feedback was based on the following assumptions, for which there is some experimental basis: (1) expression of LuxR is regulated by both LuxR–AI complexes and cAMP–CRP; (2) the stoichiometry of the LuxR·AI complex is such that two molecules of LuxR are coupled to two molecules of AI; (3) there is a non-zero, basal, AI-independent synthesis of LuxR. In the model analysis, we have examined the influence of these assumptions on the properties of the output using the following system of differential equations: In this system, the first equation describes the rate of change of the concentration of LuxR (denoted as R), which positively depends on the sum of the basal (c0) and auto-induced synthesis rates and negatively depends on constitutive degradation and dilution due to cell division. The inducible synthesis rate, described by the second term of the first equation, is assumed to be proportional to the probability of transcriptional initiation controlled by the (LuxR·AI)2 complex (C) binding to the corresponding binding site in the regulatory sequence of the operon with the dissociation constant KD. The second and third equations describe the formation of the LuxR·AI complex through formation of the intermediate bi-molecular LuxR·AI complex (RA), which can either dissociate or form a more stable ternary molecular complex C. The [AI] is denoted as A. In the model (1), we assume constant glucose concentration, leading to no regulatory role of cAMP–CRP. CRP-mediated regulation is explored in an expanded model below. For this analysis, it is important that LuxR dimerizes to form the complex C, whereas the exact stoichiometry of the complex with respect to the number of AI molecules is not consequential. The same results would be valid if C was tri-molecular, with one molecule of AI and two molecules of LuxR. At steady state, all derivatives in the systems (1) are equal to zero, which converts both systems into the same system of algebraic equations, which in turn can be reduced, with γ=k1/k2, δ=c0/k2 and β=KD/αA2, to the following form: The steady-state levels of luxR expression are obtained by solution of equation (2), which has three roots. One can further show that two of the steady states are stable and one is unstable, further implying that the system is indeed bistable. To determine the bifurcation diagram defined by (2), the method used by Ozbudak et al (2004) was applied. At the boundary between the monostable and bistable regimens, two of the three solutions to (2) coincide. Denoting them as a and denoting the third, distinct, solution as θa, one has (y−a)(y−a)(y−θa), or Comparing the coefficients in this equation with those in (2), one can obtain the following system of parametric equations for the parameters γ, δ and β: Both parameters, (γ+δ)2/β and μ=γ/δ are non-dimensional. Using (4) one can create the bifurcation diagram in Figure 3A. It is apparent from this diagram that, for certain combinations of parameters, two steady states can coexist for the expression levels of luxR, and by extension, those of GFP expression in the lux01 strain. Raising the concentration of the exogenous [AI] is equivalent to ‘moving’ on this diagram parallel to the X axis from the far left uninduced state to the far right induced state with a transition through a bistable regimen (dashed line with arrows). Importantly, for this type of transition, the ratio μ=γ/δ=k1/c0 (i.e. the ratio of the strengths of inducible and constitutive transcription) has to be greater than approximately 8. Figure 3.Mathematical modeling and experimental analysis of the lux01 circuit response. (A) The model analysis predicts the existence of a domain of non-dimensional parameters characterizing the influence of AI and cAMP–CRP (see the text for the parameter description) on the number and stability of the steady states. The variation of [AI] is equivalent to moving on this graph parallel to the X axis. The bistability curve describing the expression of luxR as a function of [AI] corresponding to the dashed line with arrows is shown in black in (B). (B) The predicted dependencies of luxR expression (R) on [AI] (proportional to ) for δ=1 and γ=20 (black), 16, 12, 8 (increasing lightness of the blue color). The bistability region corresponding to the overlap of two stable response branches fL and fH is shaded in gray. Note the progressive decrease in both the magnitude of responses and the location and the extent of the bistability regions. (C) Experimental analysis of the regions of hysteresis for glucose concentrations of 2.5 (light gray), 1.5 (medium gray) and 0 mM (dark gray, copied from Figure 1C for comparison). Red lines indicate induction from low [AIex] and blue lines indicate dilution from high [AIex]. Note the reduction of the bistability ranges and the decrease in the response amplitudes with increasing glucose, in agreement with the predictions in (B). Error bars represent the standard deviation of two independent triplicate sets. (D) Flow cytometry analysis of the hysteresis in lux01 single cell response. The results are given as histograms corresponding to different modes of response induction: ‘UP’ for the induction for 7 h from the initial [AIex]=0 nM to [AIex]=10 nM, and ‘DOWN’ for the dilution experiment from the initial [AIex]=100 nM to [AIex]=10 nM. The histograms of the ‘UP’ induction experiments for the final [AI]=100 nM and the uninduced control (0 nM) are shown for comparison. Download figure Download PowerPoint The strength of inducible transcription is regulated, in part, by the occupancy of the CRP-binding site, and thus by the level of glucose, as absence of glucose activates CRP. As the concentration of glucose increases, the occupancy of the CRP site decreases, and the bistability regimen may exist in a progressively narrower range of [AIex]. To account for this, the system (1) must be modified to explicitly include the effect of cAMP–CRP binding. Following the example of Buchler et al (2003), we can write for the equivalent of equation (3): Here, P denotes the concentration of active CRP and k7 is the equilibrium association constant of cAMP–CRP binding to its cognate-binding site. Making the substitution: η=k7P, present in equation (5), leads to: where the weight w=η/(1+η) varies between zero and unity, as CRP varies from zero to maximal values. Variation of the contribution of γ to equation (5) is equivalent to raising or lowering along the Y axis the ‘trajectory’ describing the response to AI variation in the bifurcation diagram above, with the trajectory itself being parallel to the X axis. In particular, increasing the glucose levels would be equivalent to moving the trajectory lower. Ultimately, when wγ/δ levels become less than approximately 8, bistability is lost. In an alternative model, no positive autoregulation of luxR expression is assumed. This corresponds to equating k1 to zero in (1), thus the value of γ=0. As seen above, this implies no bistability in the response. Therefore, positive LuxR autoregulation is required for a bistable response in the framework of our model. Validating model predictions: dependence of lux01 output on glucose In addition to positive autoregulation, bistability is predicted to be critically dependent on the existence of basal constitutive luxR expression, as bistability is lost when c0=0 and thus δ=0. The model also predicted that the range of [AIex] spanning the region of bistability would expand with increasing ratio of the rates of inducible to constitutive luxR transcription, γ/δ. Owing to the fact that the rate of luxR transcription is regulated in part by occupancy of the cAMP–CRP-binding site, the range of [AIex] in which LuxR bistability could occur was predicted to expand as the concentration of glucose decreased. This effect can be seen both in the bifurcation histogram in Figure 3A and in a more direct representation of bistability of R (=[LuxR]) expression as a function of , which in turn is proportional to A (=[AI]) (Figure 3B). The maximum luxR expression was predicted to rapidly diminish with increasing glucose and correspondingly decreasing [cAMP–CRP], whereas the bistability range was expected to shift to higher [AI] under the same conditions. To test the model predictions, the lux01 circuit response was analyzed at different glucose levels. Analysis of the response using the aforementioned dilution techniques in media with different glucose concentrations (0, 1.5 and 2.5 mM) confirmed the model predictions (Figure 3C). As discussed above, in RM minimal medium supplemented with succinate, hysteresis occurred in the range of 0–50 nM AI. In agreement with the model, smaller [AIex] ranges yielding bistable responses were seen when glucose concentration was increased. Moreover, the bistability ranges