Advertisement
  • Loading metrics

Both Ligand- and Cell-Specific Parameters Control Ligand Agonism in a Kinetic Model of G Protein–Coupled Receptor Signaling

  • Tamara L Kinzer-Ursem,

    ¤ Current address: Division of Biology, California Institute of Technology, Pasadena, California, United States of America

    Affiliation Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan, United States of America

  • Jennifer J Linderman

    To whom correspondence should be addressed. E-mail: linderma@umich.edu

    Affiliations Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan, United States of America , Department of Biomedical Engineering, University of Michigan, Ann Arbor, Michigan, United States of America

Both Ligand- and Cell-Specific Parameters Control Ligand Agonism in a Kinetic Model of G Protein–Coupled Receptor Signaling

  • Tamara L Kinzer-Ursem, 
  • Jennifer J Linderman
PLOS
x

Abstract

G protein–coupled receptors (GPCRs) exist in multiple dynamic states (e.g., ligand-bound, inactive, G protein–coupled) that influence G protein activation and ultimately response generation. In quantitative models of GPCR signaling that incorporate these varied states, parameter values are often uncharacterized or varied over large ranges, making identification of important parameters and signaling outcomes difficult to intuit. Here we identify the ligand- and cell-specific parameters that are important determinants of cell-response behavior in a dynamic model of GPCR signaling using parameter variation and sensitivity analysis. The character of response (i.e., positive/neutral/inverse agonism) is, not surprisingly, significantly influenced by a ligand's ability to bias the receptor into an active conformation. We also find that several cell-specific parameters, including the ratio of active to inactive receptor species, the rate constant for G protein activation, and expression levels of receptors and G proteins also dramatically influence agonism. Expressing either receptor or G protein in numbers several fold above or below endogenous levels may result in system behavior inconsistent with that measured in endogenous systems. Finally, small variations in cell-specific parameters identified by sensitivity analysis as significant determinants of response behavior are found to change ligand-induced responses from positive to negative, a phenomenon termed protean agonism. Our findings offer an explanation for protean agonism reported in β2--adrenergic and α2A-adrenergic receptor systems.

Author Summary

G protein–coupled receptors (GPCRs) are transmembrane proteins involved in physiological functions ranging from vasodilation and immune response to memory. The binding of both endogenous ligands (e.g., hormones, neurotransmitters) and exogenous ligands (e.g., pharmaceuticals) to these receptors initiates intracellular events that ultimately lead to cell responses. We describe a dynamic model for G protein activation, an immediate outcome of GPCR signaling, and use it together with efficient parameter variation and sensitivity analysis techniques to identify the key cell- and ligand-specific parameters that influence G protein activation. Our results show that although ligand-specific parameters do strongly influence cell response (either causing increases or decreases in G protein activation), cellular parameters may also dictate the magnitude and direction of G protein activation. We apply our findings to describe how protean agonism, a phenomenon in which the same ligand may induce both positive and negative responses, may result from changes in cell-specific parameters. These findings may be used to understand the molecular basis of different responses of cell types and tissues to pharmacological treatment. In addition, these methods may be applied generally to models of cellular signaling and will help guide experimental resources toward further characterization of the key parameters in these networks.

Introduction

G protein–coupled receptors (GPCRs) are the largest class of cell membrane receptors with almost 2,000 members identified [1]. It is estimated that more than 50% of pharmaceuticals target GPCRs [2]. While the majority of pharmacologic research has focused on ligand-specific properties that influence cell behavior, relatively few studies focus on cell-specific parameters that may also determine cell responses [35]. Studying the effect of changes in both ligand- and cell-specific parameters on cellular behavior is complicated by the large number of interactions and feedback mechanisms inherent in cellular signaling. Thus it becomes necessary to use quantitative models to aid in the analysis of these systems.

Typical models of GPCR signaling are termed ternary complex models or TCMs (reviewed in [68]). These models feature ligand (L) binding to receptor (R) to form a ligand–receptor complex (LR), and LR interaction with G protein (G) to form the ternary ligand–receptor–G protein (LRG) complex. Subsequent equilibrium models of GPCR signaling have remained true to this paradigm while incorporating additional receptor (e.g., active receptor R* or inactive receptor R) or G protein states or other effectors to account for experimental findings [915]. A key feature of these models is that active receptors can associate with G protein in the absence or presence of ligand to form R*G and LR*G, respectively, and both these complexes can signal. Kenakin and colleagues have proposed a thermodynamically complete representation of ligand, receptor, and G protein interactions termed the cubic ternary complex model (cTCM) and shown in Figure 1 [11]. In this model, inactive receptor can both bind ligand (to form LR) and associate with G protein (to form RG or LRG).

thumbnail
Figure 1. The Cubic Ternary Complex Model and the Cubic Ternary Complex Activation Model

The cTCM (black) is a thermodynamically complete equilibrium representation of ligand (L), receptor (R), and G protein (G) interactions [11]. Association and dissociation of L and R is represented here from top to bottom, R and G interactions from front to back, and the interconversion of inactive and active R states from left to right of the cube. Based on the cTCM, the cTCAM (black and red) incorporates the dynamics of activation and recycling of G protein (dashed lines) into a kinetic model of LRG interactions [22]. A brief summary of model parameters is found in Table 1. Description of model parameters, assumptions, and equations are given in Text S1.

https://doi.org/10.1371/journal.pcbi.0030006.g001

TCMs are typically equilibrium models and while they have been widely used, it is well known that kinetic models are better able to replicate the intrinsic dynamics of signal transduction, as has been discussed previously (see [14,1620]). Furthermore, predictions of kinetic and equilibrium models with similar parameter values can be markedly different [21,22]. Indeed, a number of groups have discussed the importance of kinetics in analyzing GPCR systems [8,20,2327]. Our group has thus developed a kinetic version of the cTCM termed the cubic ternary complex activation model (cTCAM, Figure 1) [22]. The cTCAM incorporates a G protein activation feedback loop whereby G proteins (G) couple to and are activated by active receptors (R* and LR*), allowing for GTP binding and uncoupling of the G protein heterotrimer into α and βγ subunits. The GTP on active G proteins (GαGTP) is hydrolyzed by the intrinsic GTPase activity of the alpha subunit (with or without participation of regulator of G protein signaling (RGS) proteins) to form inactive GαGDP subunits. The feedback loop is completed when the GαGDP and Gβγ subunits couple to reform inactive G protein (G). We found that the predictions of the kinetic model (cTCAM) can be strikingly different than those of the equilibrium model (cTCM) in terms of the character of the response (positive/neutral/inverse agonism), suggesting the importance of using the more realistic dynamic model [22].

In this work we use the cTCAM as a dynamic model of the initial events in GPCR signaling to identify both the ligand- and cell-specific parameters that may be key determinants of the character of cellular response. One drawback of the kinetic cTCAM, and indeed for most if not all signal transduction models, is that many parameter values have not been measured and others may vary over several orders of magnitude [10,12,22,25,28]. Additionally, the large number of parameters and incorporation of the G protein activation feedback loop make intuition of model behavior difficult. Thus it becomes necessary to introduce techniques for efficient sampling of the input parameter space and quantification of model output. In the risk analysis and environmental engineering fields, uncertainty and sensitivity analysis have been routinely used to sample input parameter space and identify key parameters [29,30]. These techniques have recently been introduced into the biological sciences. In particular, Latin hypercube sampling (LHS) with partial rank correlation coefficients (PRCC) has been used to perform uncertainty and sensitivity analysis in epidemiological studies of HIV and tuberculosis [31,32], to understand the dynamics of tuberculosis infection and immunity in host–pathogen models [33,34], and to analyze parameter sensitivity in a T cell receptor activated Erk–MAPK signaling pathway [35]. Additionally, several studies have used genetic algorithm-based search methods to perform parameter fitting and sensitivity analysis for models of T cell receptor and GPCR activation [12,35,36]. Rundell and colleagues found that these global analysis methods (LHS/PRCC and genetic algorithm approaches) and others (Sobol's method and Fournier amplitude sensitivity test (FAST)) give very similar results [35].

LHS has been shown to be a computationally efficient method for sampling parameter ranges. It is more than one order of magnitude more efficient than random sampling methods [29,30,37]. Additionally, statistical techniques can be used to identify parameters that are most important in determining output variables. Correlation coefficients can be readily calculated to identify parameters whose variation is strongly correlated with variations in an output parameter of interest. For nonlinear monotonic systems such as the cTCAM, PRCC is known to be the most appropriate [29,38,39]. PRCC values can be calculated at each time point of the simulation, and the relative importance of the parameters can be tracked over time. Here we use LHS and PRCC to identify parameters that are important determinants of G protein activation in a general model of GPCR signaling (cTCAM, Figure 1). In particular, we are interested in how small variations in parameter values might give rise to large differences in the character (positive/neutral/inverse agonism) of ligand-induced responses.

Different ligands, while binding to the same receptors, are often able to transmit different levels of signal per bound receptor and thus have different levels of response. Recent studies have shown that the same ligand may not only induce varying levels of response but also both positive and negative responses in some GPCR systems. Here we refer to the ability of a ligand to act both as a positive agonist and an inverse agonist as protean agonism (a term previously introduced by Kenakin [40]). Protean agonists (ligands that are able to induce protean agonism) have been identified in several GPCR systems, including β2-adrenergic, α2A-adrenergic, opioid, and histamine H3 receptors [4144]. After identification of ligand- and cell-specific parameters that play critical roles in determining the character of a response, we then focus on two particularly interesting studies of β2-adrenergic and α2A-adrenergic receptors in which both positive and inverse agonism are produced by introduction of the same ligand to the system. Chidiac and colleagues reported that the β2-adrenergic receptor partial agonist dichloroisoproterenol (DCI) acted as both a positive and an inverse agonist in Sf9 cells overexpressing β2-adrenergic receptor despite the fact that the treatment of the cells was similar in both cases [41]. Jansson and colleagues reported that the α2A-adrenergic ligand levomedetomidine (levomed) had opposite effects on cAMP production in different cell lines, activating the receptor in several systems (S115 and PC10 cells) while inhibiting constitutive activity of the endogenous receptor in HEL 92.1.7 cells [42,45,46]. We find that changes in cell-specific parameters identified as key determinants of response behavior by sensitivity analysis are consistent with these seemingly contradictory behaviors in the β2- and α2A-adrenergic receptor systems.

Results

Cell-Specific Parameters Are Highly Correlated with Variation in Response Characteristics

LHS was used to efficiently sample parameter values from the ranges listed in Table 1. One LHS simulation typically sampled each of the 16 parameters 1,000 times, producing 1,000 solutions to the model equations (Text S1). The formation of GαGTP is computed (see Equation S.27 in Text S1). Eight example solutions of the time course of GαGTP formation are shown in Figure 2A. As expected, variations in parameter values caused large variations in response behavior. For example, for the cases shown in Figure 2A, at steady state prior to ligand addition (time zero), GαGTP values varied between approximately two and 240 per cell. Once ligand is added to the system, it binds to receptors which in turn bind to G protein, changing the distribution of receptors and G proteins among their various states and causing increases or decreases in G protein activation (as seen by changes in GαGTP formation, Figure 2A). In some instances, G protein activation increases rapidly, and then falls due to GTP hydrolysis. Ligand-induced responses occur on the order of 5 s to 30 s at this ligand concentration, which is approximately the timescale over which G protein activation is known to occur [4749]. Calculating the percent change of GαGTP upon ligand addition (designated %OverBasal) allowed for easier observation of decreases in G protein activity upon ligand binding; in Figure 2B these are seen as negative %OverBasal values. In this way, responses were directly related to the pharmacological classifications of ligand efficacy—positive (increase in %OverBasal), neutral (no change in %OverBasal), and inverse agonism (decrease in %OverBasal)—and thus can be compared with experimental data that normalize to control conditions.

thumbnail
Figure 2. Time Course of Representative Model Outputs

Parameter values are sampled using LHS, and the differential equations describing the cTCAM are solved according to the equations in Text S1. G protein activation as quantified by GαGTP is tracked over time (see Equation S.27 in Text S1).

(A) Values of GαGTP (number/cell) for eight parameter sets from LHS sampling of the ranges in Table 1 are plotted over the course of the simulation.

(B) Percent change in the value of GαGTP relative to basal values (%OverBasal) was calculated and tracked over time according to Equation 1. [L] = 0.1 μM.

https://doi.org/10.1371/journal.pcbi.0030006.g002

To determine the correlation between parameter values and levels of G protein activation, PRCC values were calculated at 0.25-s intervals for varying ligand concentrations (0.1 nM to 0.1 mM). Correlations (PRCC values) for each parameter listed in Table 1 were calculated with respect to the two different measurements (outputs) of G protein activation discussed above, the number of GαGTP and %OverBasal. Table 2 shows the rank order of PRCC values for the two responses at two time points, 5 s and 2.5 min, and two ligand concentrations, 1 nM and 10 μM, representing pre- and post-steady state time points and sub- and supra-saturating ligand conditions, respectively.

Previous analysis of the equilibrium cTCM in our group found that Kact, Gtotal, α, δ, γ, and Kg all played a role in determining the character (positive/neutral/inverse agonism) of the response [20]. By using the cTCAM to include the necessary activation events, however, a somewhat different set of parameters was found to be highly correlated with response generation as summarized in Table 2. Not surprisingly, the ligand-specific parameter most correlated with both measures of response generation (instantaneous number of GαGTP and %OverBasal) was the effectiveness with which the ligand induces an active receptor conformation (α). Indeed, this was the only ligand-specific parameter found to be highly correlated with the GαGTP response. The character of responses was influenced by several cell-specific parameters including receptor and G protein expression (Rtotal and Gtotal), parameters involved in the G protein activation loop (kGact, kGTP, and kG), the equilibrium ratio of active to inactive receptor (Kact), and the rate and efficiency of receptor–G protein coupling (β and k11). Although generally the parameters that were highly correlated did not differ between the two response measures (GαGTP and %OverBasal), the rank order of the PRCC values did vary. Similar results were found when the microscopic reversibility assumption of the model (discussed in the Methods section) was relaxed and all forward and reverse rate constants (shown in Figure S1) were varied (unpublished data).

The rank order of the parameters that were highly correlated with GαGTP did not vary significantly with time, and thus PRCC values for the parameters at 2.5 min but not 5 s are shown in Table 2. Additionally, the PRCC values did not vary significantly at varying ligand concentrations, with the exception that kG and α were found to be lower in rank order at the lower ligand concentrations.

For the %OverBasal response, significant differences were seen between low and high ligand concentrations. At low ligand concentration (1 nM in Table 2), the parameters for reversible ligand binding (k3 and Ka) were highly correlated with %OverBasal. Additionally, the PRCC values of both k3 and Ka were found to vary with time, and thus the PRCC values for all parameters at both 5 s and 2.5 min at this ligand concentration are listed in Table 2. As shown in Figure 3, the ligand association rate constant k3 was found to be highly correlated at 5 s, but was not significantly correlated after 2.5 min of ligand binding. In contrast, the PRCC value for Ka rapidly increased over the first 30 s of the simulation and was significantly correlated with %OverBasal at longer times (greater than 1 min). Thus at sub-saturating ligand concentrations in the first few seconds of ligand binding when response characteristics are likely to be determined, changes in response are highly sensitive to the ligand association rate constant. At high ligand concentration (10 μM in Table 2), k3 and Ka were not highly correlated with %OverBasal, nor was there a significant difference between PRCC values at the two times, and thus only PRCC values at 2.5 min are listed in Table 2.

thumbnail
Figure 3. Example of the Time Course of PRCC Values

Time course of PRCC values for LR association rate constant (k3) and LR equilibrium association constant (Ka) correlated to %OverBasal (as given by Equation 1) when [L] = 1 nM.

https://doi.org/10.1371/journal.pcbi.0030006.g003

Receptor and G Protein Levels Independently Affect Response Characteristics

The levels of receptor and G protein expression were both identified as important parameters in response generation (Table 2). To investigate the impact that variations in these parameters have on the character of response generation, simulations varying Rtotal and Gtotal over their physiologic range were performed. To simulate (but not replicate) experiments that measure the accumulation of a signal over time, such as radioligand assays, the integral of GαGTP over the first 10 s of ligand binding was calculated and normalized to basal values with no ligand present and is designated %Accum according to Equation 2. Qualitatively similar results are obtained using the instantaneous number of GαGTP at either 5 s or 10 s after ligand binding (unpublished data). For certain sets of parameters values, particularly when α, δ, and γ ≠ 1, the %Accum upon ligand addition may vary from positive to negative values. The explanation for this protean agonist behavior is as follows. In conditions of constitutive activity (L = 0, Kact ≠ 0), there exist two receptor–G protein complexes (RG and R*G) of which only R*G can activate G protein. Upon ligand addition, the number of receptor–G protein complexes increases to four (RG, R*G, LRG, and LR*G), two of which (R*G and LR*G) can activate G protein. At low values of Rtotal and Gtotal, the addition of ligand to the system can actually reduce G protein activation because of the redistribution of receptors and G proteins among their various states. When this occurs, the ligand behaves as an inverse agonist; a decrease in the %Accum upon ligand addition is then seen (Figure 4A and 4B). As the total numbers of receptors and G proteins are increased, the total number of R*G and LR*G generated upon ligand addition greatly outnumber those of R*G prior to ligand addition, and the ligand behaves as a positive agonist [50]. Figure 4A shows the behavior of %Accum when Rtotal is held constant at 5,000/cell and Gtotal is varied. Note that as G protein expression is increased, the same ligand is predicted to change from an inverse to a neutral to a positive agonist. Similar trends are seen when Gtotal is held constant at 10,000/cell and Rtotal is varied (Figure 4B).

thumbnail
Figure 4. The Effect of Changing Receptor and G Protein Expression on the Activation of G Protein

Dose response curves measuring the percent accumulation of GαGTP (%Accum) were calculated according to Equation 2 as described in Methods.

(A) Total G protein (Gtotal) is varied from 1,000,000/cell to 3,000 per cell, and Rtotal = 5,000/cell.

(B) Total receptor (Rtotal) is varied from 30,000 to 1,000 per cell, and Gtotal = 10,000/cell. Parameter values: k1 = 1 s−1, k3 = 1 × 107 M−1s−1, k11 =1 × 10−4 (number/cell)−1s−1, kGact = 5 s−1, kGTP = 1 s−1, kG = 1 × 10−4 (number/cell)−1s−1, Ka = 1 × 10−8 M−1, Kg = 1 × 10−4 (number/cell)−1, Kact = 0.01, α = 5, β = 5, δ = 0.5, γ = 0.1, η = 0.1.

https://doi.org/10.1371/journal.pcbi.0030006.g004

Previous studies have proposed that the ratio of receptors to G proteins may play an important role in determining the efficacy of a ligand [40,51]. Furthermore, previous analysis of the equilibrium cTCM has shown that relatively large changes (>25-fold) in Kact and Gtotal may change the efficacy of a ligand, from positive to neutral or neutral to negative responses [50]. To investigate the role of the R/G ratio in our more physiological model, %Accum at a saturating ligand concentration (10 μM) from Figure 4A and 4B was plotted versus the ratio of Rtotal/Gtotal in Figure 5. As seen in Figure 4, the level of response increased as either receptor or G protein levels were increased. However, because increasing Gtotal decreases the ratio R/G, the response generated in simulations of constant Rtotal and varying Gtotal (dashed line in Figure 5) increases as Rtotal/Gtotal decreases. In contrast, as receptor expression is increased, the level of response and the ratio Rtotal/Gtotal increase as indicated by a positive slope in the curve (solid line in Figure 5). Thus, there is not a clear relationship between the ratio Rtotal/Gtotal, and this measure cannot be used as a straightforward predictor of response efficacy. These results indicate that the individual levels of receptor and G protein expression are important in determining ligand efficacy.

thumbnail
Figure 5. The Effect of the Ratio of Receptor to G Protein on G Protein Activation

Rtotal was set at 5,000/cell and Gtotal varied (dashed line). As G protein expression increases, the response changes from negative to positive agonism. Gtotal was set at 100,000 per cell and Rtotal varied (solid line). As receptor expression increases, the response changes from negative to positive agonism.

https://doi.org/10.1371/journal.pcbi.0030006.g005

Our Results Offer New Explanations for Protean Agonism in the α2A- and β2-Adrenergic Receptor Systems

α2A-Adrenergic receptors couple to Gαi proteins, activating the G proteins, which in turn inhibit adenylyl cyclase activation. Jansson and colleagues have reported that the α2A-adrenergic ligand levomed is a positive agonist in PC10 cells, causing an inhibition of cAMP production as shown in Figure 6A [45]. However, the same group has reported that levomed also acts as an inverse agonist in HEL 92.1.7 cells, causing an increase in cAMP production (Figure 6A) [42]. These results suggest that a parameter (or parameters) different between the two cell types may play a critical role in determining the character of the response, and cause protean agonism.

thumbnail
Figure 6. Protean Agonism in the α2A-Adrenergic System

(A) Effect of levomed on cAMP production. Note that the y-axis in this plot is inverted from the usual to show positive agonists to have a positive slope and inverse agonists to have a negative slope. Levomed acts as an inverse agonist in HEL 92.1.7 cells (•, with curve fit). Data taken from Jansson et al. (1998), Figure 4. Levomed acts as a positive agonist in PC12 cells (line only). Data reconstructed from EC50 and max percent inhibition reported in Jansson et al. (1994), Table 1.

(B–D) Simulations of protean agonism of levomed at the α2A-adrenergic receptor. Small changes in parameter values can cause the response to switch from positive to negative.

(B) 3.3-Fold variation in G protein expression, β = 10.

(C) A 4-fold variation in the G protein activation rate constant kGact, Gtotal = 100,000.

(D) The equilibrium ratio of active to inactive receptors is varied 5-fold, Gtotal = 10,000, kGact = 5 s−1. Parameter values are equal to those listed in Figure 5 except when otherwise noted. Rtotal = 3,500 number/cell. Simulated dose response curves (B–D) measuring the percent accumulation of GαGTP (%Accum) were calculated according to Equation 2 as described in Methods.

https://doi.org/10.1371/journal.pcbi.0030006.g006

Guided by the results from our uncertainty and sensitivity analysis (Table 2), we tested whether changes in a single cell–specific parameter, a parameter that might differ between the PC10 and HEL 92.1.7 cells, could produce protean agonism. Small (less than or equal to half an order of magnitude) changes in any of three parameters, Gtotal, kGact, or Kact, were all able to produce protean agonism, as shown with representative simulations in Figure 6B–6D. All three changes represent physiologically reasonable explanations for the protean agonism. Differences in G protein expression (Gtotal) between two cell lines and indeed within a cell line are common, although actual expression levels are rarely quantified [5254]. Differences in the rate constant of G protein activation (kGact) between cell lines then may be due to differences in G protein isoform expression profiles [55]. Further, differences in the equilibrium ratio of active to inactive receptors (Kact) between cells expressing endogenous receptors and those with transfected receptors, in this case, between the transfected PC10 and endogenous HEL 92.1.7 cells studied by Jansson and colleagues, would also seem quite plausible.

Interestingly, although differences in α2A-adrenergic receptor expression between HEL 92.1.7 cells (2,900–4,100 receptors at the cell membrane; [56]) and PC12 cells (about 5-fold greater; [57]) have been reported, in our simulations these modest changes in Rtotal did not produce protean agonism. Similarly, although the GTPase activity of the Gα protein subunit (kGTP) can vary widely between different cell types depending on the expression of RGS proteins [3,25,58], variation of kGTP did not produce protean agonism in our model. Other cellular parameters identified by uncertainty and sensitivity analysis (Table 2) also did not produce protean agonism, at least not for moderate (less than an order of magnitude) changes in their values and for physiologically reasonable values of the remaining parameters.

As a second example of protean agonism, Chidiac and colleagues have reported that the β2-adrenergic ligand DCI produces both stimulatory and inhibitory responses in Sf9 cells despite similar treatment of the cells (Figure 7A) [41]. The parameter with the most likely variation in this system is G protein expression (Gtotal). While quantitative measurement of Gαs-like proteins in Sf9 cells has not to our knowledge been reported, several studies have shown that there may be wide variations in G protein expression in this system. Seifert et al. [52] and several references therein report that endogenous Gαs-like proteins could not be detected with immunoblot assays of Sf9 membranes, while Kleymann et al. [59] and Leopoldt et al. [54] have detected endogenous Gαs-like proteins in immunoblots of membranes. Additionally, infection of Sf9 cells with baculoviruses has been shown to downregulate the expression of endogenous G proteins [54]. Simulations at varying Gtotal showed that as little as a 5-fold change in Gtotal was able to produce protean agonism in our model, as shown in Figure 7B.

thumbnail
Figure 7. Protean Agonism in the β2-Adrenergic Receptor System

(A) Dichloroisoproterenol (DCI) effect on adenylyl cyclase activity in Sf9 cells. DCI was found to be both a partial agonist (•) and an inverse agonist (○) in this study. Data replotted from Chidiac et al. (1996).

(B) Simulations of DCI activation of GαGTP. Protean agonism properties of DCI caused by 5-fold difference in G protein concentration.

(C) After desensitizing treatment with isopreterenol, DCI was found to inhibit adenylyl cyclase activity in membranes where previously positive agonism was seen (•). This treatment further decreased activity of adenylyl cyclase in membranes where inverse agonism was observed (○). Data replotted from Chidiac et al. (1996).

(D) Desensitization treatment by isopreterenol is simulated by decreasing G protein (Gtotal) by 50%. Parameter values are equal to those listed in Figure 4 except when otherwise noted. Rtotal = 4,000/cell, α = 0.5, δ = 5. Simulated dose response curves (B,D) measuring the percent accumulation of GαGTP (%Accum) were calculated according to Equation 2 as described in Methods.

https://doi.org/10.1371/journal.pcbi.0030006.g007

Chidiac and colleagues further report that after treatment with isoproterenol (a β2-adrenergic agonist) DCI acts as an inverse agonist only (Figure 7C) [41]. In other words, protean agonism by DCI is not seen after receptor desensitization. In another cell type (S49 lymphoma cells), Insel and colleagues have found that redistribution of Gα protein subunit between the membrane and cytosolic compartments occurs after prolonged treatment with isoproterenol [60]. The possibility that this loss of G proteins contributes to inverse agonism in Sf9 cells was tested by decreasing the number of available G proteins (Gtotal). Ligand stimulation of these desensitized cells (parameters as in Figure 7B but with half the number of G proteins) produced only inverse agonism in our model (Figure 7D). Thus our analysis suggests the observation of positive or inverse agonism in these cells is sensitive to G protein number and that small changes in G protein number can account for observations of both protean agonism (before desensitization) and inverse agonism (after desensitization).

Discussion

As is common in large signal transduction networks, there are significant uncertainties in model parameter values, and it is difficult to intuit model behavior. Using efficient parameter variation and sensitivity analysis methods, we identify parameters in a dynamic G protein activation model that play key roles in determining the level and indeed the character (positive/neutral/inverse agonism) of the response. This type of analysis allows for quick and efficient identification of important parameters. Drug design and development typically focus on altering ligand-specific parameters to produce desired cell response. Significantly, however, we find that not only ligand-specific parameters, but also cell-specific parameters, play critical roles in determining response behavior. To demonstrate how changes in cellular specific parameters can dramatically change cell response, we apply our findings towards studies of protean agonism in the α2A- and β2-adrenergic receptor systems.

Our analysis shows that several cell-specific parameters not previously identified contribute significantly to the character of ligand-induced responses. These include the total receptor concentration (Rtotal), the GTP hydrolysis rate constant (kGTP), and the G protein activation rate constant (kGact) (see Table 2 for a full list). Receptor number is known to vary widely in different cell types, and is regularly manipulated using transfection technologies. This is one of the few parameters in our model that is routinely quantified. It is well known that the GTPase activity of the Gα protein subunit may vary and is largely dependent on the presence of RGS proteins in the system [61,62]. Therefore, it is not surprising that the rate constant for GTP hydrolysis (kGTP) would be found to be important in response generation; indeed much attention has been given to RGS proteins as new therapeutic treatments [3,63,64]. Our findings are also consistent with an elegant study by Bornheimer and colleagues that analyzed G protein activation by active receptor and deactivation by GTPase activation proteins (GAPs) [12]. They found that local concentrations of receptors and GTPase activation proteins mediate various regimes of response behavior by kinetically controlling G protein activity. Following its identification by sensitivity analysis, we find that small changes in kGact (the rate constant for G protein activation) may result in protean agonism such as reported by Jansson and colleagues for the α2A-adrenergic system [42,45].

G protein concentration and Kact, the ratio of active to inactive receptor states, are two cell-dependent parameters identified by our study (using a dynamic model of G protein activation) and two previous studies (using equilibrium models) as parameters that are key to determining the character of response and, when varied, could cause protean agonism (Table 2 and Figures 6B, 6D, and 7B) [20,40,50]. Overexpression or underexpression of receptor and G protein or expression of different isoforms of G proteins may give very different results than those found in endogenous systems. Additionally, G protein expression may vary largely in cells depending on a variety of factors, including cell type, state of cell development [65], prior treatments to the cells [54,60,66,67], and disease state [6870]. Although Kact has not been rigorously quantified, receptor mutation and fluorescent imaging studies are lending valuable insight into how conformational changes in the receptor confer activity with and without ligand stimulus [7175].

Our findings on the sensitivity of G protein activation to these parameters are corroborated by experimental studies implicating changes in G protein and receptor expression and receptor activity in cardiac and Alzheimer disease. For example, it has been shown that both β1- and β2-adrenergic receptors are expressed at decreased levels in studies of cardiac failure, while Gαi subunits are increased [76]. As a second example, although muscarinic receptor density is not changed in brains of Alzheimer patients, the functionality of the receptors has been shown to be compromised (suggesting an inactive receptor state and changes in Kact), and decreases in the function of Gαq protein have also been reported [77]. Thus, both experimental reports and our modeling results suggest that small changes in these key parameters may disrupt normal signaling and lead to disease states.

To account for day-to-day and cell-to-cell variability, experimental results are almost always presented normalized to basal levels. However, clearly some information is lost when normalizing, both in experimental data and in modeling studies. For example, in experimental systems normalizing washes out variations in the “basal state” of the cells that may be an indicator of the current state of the cell or of future response characteristics. In the context of our model, un-normalized cell population or single cell data would allow for a more in-depth study of how parameters contribute to both basal and post-ligand treatment responses.

Our findings represent what may be only a small sampling of cellular parameters that influence ligand efficacy. Indeed, small variations in combinations of parameters could also lead to observations of protean agonism. For the sake of conciseness, these are not analyzed here. However, the methods introduced here provide a computationally efficient mechanism to begin to explore all possibilities. One limitation of this type of analysis, particularly PRCC analysis, is that it can only identify trends in certain directions within the given parameter space. Other analysis techniques for LHS sampling, such as subjective, differential sensitivity analysis, one-at-a-time design, and the adjoint method have been used, although these also have significant limitations [39]. Another limitation for using PRCC is that the system must be monotonic; however, this is easily checked by monitoring scatterplots, and there exist other global analysis methods by which to analyze the impact of parameter variation on model output. Recent work has compared multiple global analysis methods and found that LHS/PRCC, genetic algorithm approaches, Sobol's method, and Fournier amplitude sensitivity test (FAST) give very similar results [35].

Finally, both kinetic measurements and modeling will be important to our progress in understanding and manipulation of G protein activation. New technologies, such as those reported in yeast and HEK 293 cells using fluorescence resonance energy transfer (FRET) and bioluminescence resonance energy transfer (BRET) to monitor receptor–G protein and G protein subunit interactions [48,49] have the potential to quantify these interactions in real time. Modeling studies by our group and others [12,33,7883] facilitate our understanding of the complexities involved in cell signaling and identify pathway interactions that are key to describing normal and pathological cell functions. The dynamic model that we present here, a general model of GPCR and G protein activation, can be easily expanded to include details of particular GPCR systems and phenomena such as receptor desensitization and internalization, processes that do not operate under steady state conditions. Further, as the components involved in signal transduction become better described and incorporated into signaling databases (e.g., SigPath [84], Alliance for Cell signaling [85], National Center for Genome Resources's PathDB [86]) and as more complex models of signaling pathways become possible, it will become increasingly important to systematically assess parameter uncertainty and quantify regimes of model behavior.

Materials and Methods

The cubic ternary complex activation model.

The cubic ternary complex activation model (cTCAM) (Figure 1) is a kinetic extension of the (equilibrium) cTCM integrating a feedback loop of G protein activation and recycling with dynamic LRG interactions [22]. Each reaction in the model is governed by mass action kinetics. Reaction rate constants describe the association and dissociation of ligand and receptor (top to bottom of the cube) and receptor and G protein (front to back of the cube) as seen in Figure 1. The interconversion of inactive and active receptor states is described by forward and backward rate constants as viewed from left to right of the cube.

The cTCAM has a total of 27 kinetic rate constants, 24 describing the binding and dissociation reactions between ligand, receptor, and G protein, and three describing the G protein activation cycle. Additionally, the total concentrations of ligand, receptor, and G protein are required, bringing the total number of model parameters to 30. As described in Text S1, the model can be reduced to 16 input parameters (plus ligand concentration). A description of all the parameters used in the cTCAM is presented in Table 1. Briefly, four of these are ligand-specific parameters that control the binding of ligand to the receptor and the ability of ligand to bias receptor activation and G protein association (Ka, k3, α, and γ, respectively). The eleven cell-specific parameters in the model determine the strength of association of the inactive and active receptor states with G protein (Kg, k11, η, and β), the ratio of active to inactive receptor (k1, Kact), the kinetics of G protein activation, deactivation, and recombination (kGact, kGTP, and kG), and the numbers of G proteins and receptors (Gtotal, Rtotal). The remaining constant, δ, is both a ligand- and cell-specific parameter that governs the synergistic effects between ligand binding, receptor activation, and G protein association [11].

The equations describing the cTCAM are presented in Text S1. To calculate the initial conditions for each species in the model under varying parameter values, the total receptor and G protein concentrations (Rtotal and Gtotal) were set as the initial values of R and G while the initial values of all other species were set to zero. The system was allowed to come to steady state (typically less than 100 s) at which point a bolus of ligand (L) was added to the system (denoted as time = 0), and the formation of each species was tracked over time.

Model analysis.

The level of GαGTP is the key output of the model. However, experimental data are routinely normalized to basal values (i.e., the amount of activated G protein in the absence of ligand). Therefore, for the model G protein activation was quantified as the percent change in GαGTP from basal (L = 0) as given by Quantifying response in this way is directly related to the pharmacological classifications of ligand efficacy—positive (increase in %OverBasal), neutral (no change in %OverBasal), and inverse agonism (decrease in %OverBasal)—and is analogous to that previously used to analyze response activation in the cTCM and cTCAM models [20,22,81].

Dose response curves were generated by calculating the integral of GαGTP (∫GαGTP[t] dt) over the first 10 s of ligand stimulation at varying ligand concentrations. These dose-response curves are used to look at how responses may change upon varying receptor and G protein totals (Figure 4) and for comparison with experiments that measure the accumulation of radioligand, for example [3H]cAMP accumulation in studies of protean agonism in β2-adrenergic and α2A-adrenergic receptors (Figures 6 and 7). Values of the integral were normalized to basal values according to To distinguish this measurement from %OverBasal, this analysis is termed “%Accum.”

Uncertainty and sensitivity analysis using Latin hypercube sampling and partial rank correlation coefficient.

To efficiently sample the ranges over which input parameters (xi, i = 1,2,..,X) may vary, LHS was implemented for the cTCAM using methods described previously [29,30,87,88]. Briefly, each input parameter is assigned a range according to values found in the literature (Table 1). Each parameter's value range was divided into N equal probable segments according to a specified probability distribution function for that parameter. For a typical simulation, N = 1,000. Distributions for most of these parameters are not known and therefore uniform distributions were used for each parameter. A random value was then chosen from each segment, so that each parameter became a vector of N values. The N values of the parameter vectors were then randomly paired to generate an N by X input matrix where X was the number of parameters to be varied (X = 16 for the cTCAM). The differential equations describing the model (Text S1) were then solved, generating a vector of N solutions.

PRCCs were calculated to quantify the relative importance of each parameter in generating a desired output (measures of GαGTP described above). Partial correlation measures the strength of the linear relationship between the output and an input variable (xi) after the effect of all other elements of x have been removed [38], while the rank transformation is used to linearize the nonlinear monotonic relationship between the input parameters and the output [29]. PRCC values vary between −1 (perfect negative correlation) and 1 (perfect positive correlation). PRCC values were calculated as described previously [29,30,87,88]. Briefly, solutions of GαGTP at desired time points were added to the LHS input matrix to generate an N by X + 1 matrix. The values for each of the X + 1 parameters were then ranked from 1 to N and the resulting matrix was used to calculate a partialized matrix in which the linearized effects of the other parameters are taken out of each parameter. Correlation coefficients were then calculated from the partialized matrix. Scatterplots were generated to assure that the monotonicity assumption applies [29]. The calculated PRCC values were differentiated based on p-values derived from a Student's t test and were then ranked according to their absolute value.

Supporting Information

Figure S1.

The Cubic Ternary Complex Activation Model with Rate Constants

https://doi.org/10.1371/journal.pcbi.0030006.sg001

(77 KB TIF)

Text S1.

Cubic Ternary Complex Activation Model

https://doi.org/10.1371/journal.pcbi.0030006.sd001

(93 KB DOC)

Accession Numbers

The Swiss-Prot (http://ca.expasy.org) accession numbers for the proteins mentioned in the text are: α2a-adrenergic receptor (P08913), β2-adrenergic receptor (Q6GMT4), and Gα protein (Q6B6N3).

Acknowledgments

The authors thank Stewart T. Chang for helpful discussions of and assistance with parameter variation and sensitivity analysis methods.

Author Contributions

TLKU conceived and designed the experiments, performed the experiments, analyzed the data, contributed reagents/materials/analysis tools, and wrote the paper. JJL was the principal investigator, and analyzed data as well as wrote the paper.

References

  1. 1. Ji TH, Grossmann M, Ji I (1998) G protein–coupled receptors: I. Diversity of receptor–ligand interactions. J Biol Chem 273: 17299–17302.TH JiM. GrossmannI. Ji1998G protein–coupled receptors: I. Diversity of receptor–ligand interactions.J Biol Chem2731729917302
  2. 2. Sautel M, Milligan G (2000) Molecular manipulation of G protein–coupled receptors: A new avenue into drug discovery. Curr Med Chem 7: 889–896.M. SautelG. Milligan2000Molecular manipulation of G protein–coupled receptors: A new avenue into drug discovery.Curr Med Chem7889896
  3. 3. Neubig RR, Siderovski DR (2002) Regulators of G protein signalling as new central nervous system drug targets. Nat Rev Drug Discov 1: 187–197.RR NeubigDR Siderovski2002Regulators of G protein signalling as new central nervous system drug targets.Nat Rev Drug Discov1187197
  4. 4. Milligan G, Bond RA (1997) Inverse agonism and the regulation of receptor number. Trends Pharmacol Sci 18: 468–474.G. MilliganRA Bond1997Inverse agonism and the regulation of receptor number.Trends Pharmacol Sci18468474
  5. 5. Kenakin T (2005) New concepts in drug discovery: Collateral efficacy and permissive antagonism. Nat Rev Drug Discov 4: 919–927.T. Kenakin2005New concepts in drug discovery: Collateral efficacy and permissive antagonism.Nat Rev Drug Discov4919927
  6. 6. Cuatrecasas P (1974) Membrane receptors. Annu Rev Biochem 43: 169–214.P. Cuatrecasas1974Membrane receptors.Annu Rev Biochem43169214
  7. 7. Kenakin T (1996) Receptor conformational induction versus selection: All part of the same energy landscape. Trends Pharmacol Sci 17: 190–191.T. Kenakin1996Receptor conformational induction versus selection: All part of the same energy landscape.Trends Pharmacol Sci17190191
  8. 8. Lauffenburger DA, Linderman JJ (1993) Receptors: Models for binding, trafficking, and signaling. New York: Oxford University Press. 365 p.DA LauffenburgerJJ Linderman1993Receptors: Models for binding, trafficking, and signalingNew YorkOxford University Press365
  9. 9. De Lean A, Stadel J, Lefkowitz R (1980) A ternary complex model explains the agonist-specific binding properties of the adenylate cyclase–coupled beta-adrenergic receptor. J Biol Chem 255: 7108–7117.A. De LeanJ. StadelR. Lefkowitz1980A ternary complex model explains the agonist-specific binding properties of the adenylate cyclase–coupled beta-adrenergic receptor.J Biol Chem25571087117
  10. 10. Samama P, Cotecchia S, Costa T, Lefkowitz RJ (1993) A mutation-induced activated state of the beta 2–adrenergic receptor. Extending the ternary complex model. J Biol Chem 268: 4625–4636.P. SamamaS. CotecchiaT. CostaRJ Lefkowitz1993A mutation-induced activated state of the beta 2–adrenergic receptor. Extending the ternary complex model.J Biol Chem26846254636
  11. 11. Weiss JM, Morgan PH, Lutz MW, Kenakin TP (1996) The cubic ternary complex receptor–occupancy model I. Model description. J Theor Biol 178: 151–167.JM WeissPH MorganMW LutzTP Kenakin1996The cubic ternary complex receptor–occupancy model I. Model description.J Theor Biol178151167
  12. 12. Bornheimer SJ, Maurya MR, Farquhar MG, Subramaniam S (2004) Computational modeling reveals how interplay between components of a GTPase-cycle module regulates signal transduction. Proc Natl Acad Sci U S A 101: 15899–15904.SJ BornheimerMR MauryaMG FarquharS. Subramaniam2004Computational modeling reveals how interplay between components of a GTPase-cycle module regulates signal transduction.Proc Natl Acad Sci U S A1011589915904
  13. 13. Leff P, Scaramellini C, Law C, McKechnie K (1997) A three-state receptor model of agonist action. Trends Pharmacol Sci 18: 355–362.P. LeffC. ScaramelliniC. LawK. McKechnie1997A three-state receptor model of agonist action.Trends Pharmacol Sci18355362
  14. 14. Kukkonen JP, Nasman J, Akerman KEO (2001) Modeling of promiscuous receptor–Gi/Gs protein coupling and effector response. Trends Pharmacol Sci 22: 616–622.JP KukkonenJ. NasmanKEO Akerman2001Modeling of promiscuous receptor–Gi/Gs protein coupling and effector response.Trends Pharmacol Sci22616622
  15. 15. Kukkonen JP (2004) Explicit formulation of different receptor–G protein interactions and effector regulation. Bioinformatics 20: 2411–2420.JP Kukkonen2004Explicit formulation of different receptor–G protein interactions and effector regulation.Bioinformatics2024112420
  16. 16. Asthagari AR, Lauffenburger DA (2000) Bioengineering model of cell signaling. Annu Rev Biomed Eng 2: 31–53.AR AsthagariDA Lauffenburger2000Bioengineering model of cell signaling.Annu Rev Biomed Eng23153
  17. 17. Eungdamrong NJ, Iyengar R (2004) Computational approaches for modeling regulatory cellular networks. Trends Cell Biol 14: 661–669.NJ EungdamrongR. Iyengar2004Computational approaches for modeling regulatory cellular networks.Trends Cell Biol14661669
  18. 18. Weng G, Bhalla US, Iyengar R (1999) Complexity in biological signaling systems. Science 284: 92–96.G. WengUS BhallaR. Iyengar1999Complexity in biological signaling systems.Science2849296
  19. 19. Normile D (1999) Complex systems: Building working cells in “silico.”. Science 284: 80–81.D. Normile1999Complex systems: Building working cells in “silico.”.Science2848081
  20. 20. Woolf PJ, Linderman JJ (2000) From the static to the dynamic: 3 models of signal transduction in G protein–coupled receptors. In: Christopoulos A, editor. Biomedical applications of computer modeling. New York: CRC Press. PJ WoolfJJ Linderman2000From the static to the dynamic: 3 models of signal transduction in G protein–coupled receptors.In:. A. ChristopoulosBiomedical applications of computer modelingNew YorkCRC Press
  21. 21. Wyman J (1975) The turning wheel: A study in steady states. Proc Natl Acad Sci U S A 72: 3983–3987.J. Wyman1975The turning wheel: A study in steady states.Proc Natl Acad Sci U S A7239833987
  22. 22. Shea LD, Neubig RR, Linderman JJ (2000) Timing is everything—The role of kinetics in G protein activation. Life Sci 68: 647–658.LD SheaRR NeubigJJ Linderman2000Timing is everything—The role of kinetics in G protein activation.Life Sci68647658
  23. 23. Stickle D, Barber R (1992) The encounter coupling model for beta-adrenergic receptor/GTP-binding protein interaction in the S49 cell. Calculation of the encounter frequency. Biochem Pharmacol 43: 2015–2028.D. StickleR. Barber1992The encounter coupling model for beta-adrenergic receptor/GTP-binding protein interaction in the S49 cell. Calculation of the encounter frequency.Biochem Pharmacol4320152028
  24. 24. Thomsen WJ, Neubig RR (1989) Rapid kinetics of alpha 2–adrenergic inhibition of adenylate cyclase. Evidence for a distal rate–limiting step. Biochemistry 28: 8778–8786.WJ ThomsenRR Neubig1989Rapid kinetics of alpha 2–adrenergic inhibition of adenylate cyclase. Evidence for a distal rate–limiting step.Biochemistry2887788786
  25. 25. Zhong H, Wade SM, Woolf PJ, Linderman JJ, Traynor JR, et al. (2002) A spatial focusing model for G protein signals: RGS protein-mediated kinetic scaffolding. J Biol Chem 278: 7278–7284.H. ZhongSM WadePJ WoolfJJ LindermanJR Traynor2002A spatial focusing model for G protein signals: RGS protein-mediated kinetic scaffolding.J Biol Chem27872787284
  26. 26. Waller A, Sutton KL, Kinzer-Ursem TL, Absood A, Traynor JR, et al. (2004) Receptor binding kinetics and cellular responses of six N-formyl peptide agonists in human neutrophils. Biochemistry 43: 8204–8216.A. WallerKL SuttonTL Kinzer-UrsemA. AbsoodJR Traynor2004Receptor binding kinetics and cellular responses of six N-formyl peptide agonists in human neutrophils.Biochemistry4382048216
  27. 27. Sklar LA, Omann GM (1990) Kinetics and amplification in neutrophil activation and adaptation. Semin Cell Biol 1: 115–123.LA SklarGM Omann1990Kinetics and amplification in neutrophil activation and adaptation.Semin Cell Biol1115123
  28. 28. Weiss A, Schlessinger J (1998) Switching signals on or off by receptor dimerization. Cell 94: 277–280.A. WeissJ. Schlessinger1998Switching signals on or off by receptor dimerization.Cell94277280
  29. 29. Helton JC, Davis FJ (2002) Illustration of sampling-based methods for uncertainty and sensitivity analysis. Risk Anal 22: 591–622.JC HeltonFJ Davis2002Illustration of sampling-based methods for uncertainty and sensitivity analysis.Risk Anal22591622
  30. 30. Iman RL, Helton JC (1988) An investigation of uncertainty and sensitivity analysis techniques for computer models. Risk Anal 8: 71–90.RL ImanJC Helton1988An investigation of uncertainty and sensitivity analysis techniques for computer models.Risk Anal87190
  31. 31. Blower SM, Gershengorn HB, Grant RM (2000) A tale of two futures: HIV and antiretroviral therapy in San Francisco. Science 287: 650–654.SM BlowerHB GershengornRM Grant2000A tale of two futures: HIV and antiretroviral therapy in San Francisco.Science287650654
  32. 32. Tanaka MM, Small PM, Salamon H, Feldman MW (2000) The dynamics of repeated elements: Applications to the epidemiology of tuberculosis. Proc Natl Acad Sci U S A 97: 3532–3537.MM TanakaPM SmallH. SalamonMW Feldman2000The dynamics of repeated elements: Applications to the epidemiology of tuberculosis.Proc Natl Acad Sci U S A9735323537
  33. 33. Chang ST, Linderman JJ, Kirschner DE (2005) Multiple mechanisms allow mycobacterium tuberculosis to continuously inhibit MHC class II–mediated antigen presentation by macrophages. Proc Natl Acad Sci U S A 102: 4530–4535.ST ChangJJ LindermanDE Kirschner2005Multiple mechanisms allow mycobacterium tuberculosis to continuously inhibit MHC class II–mediated antigen presentation by macrophages.Proc Natl Acad Sci U S A10245304535
  34. 34. Marino S, Kirschner DE (2004) The human immune response to myocobacterium tuberculosis in lung and lymph node. J Theor Biol 227: 463–486.S. MarinoDE Kirschner2004The human immune response to myocobacterium tuberculosis in lung and lymph node.J Theor Biol227463486
  35. 35. Zheng Y, Rundell A (2006) Comparative study of parameter sensitivity analyses of the TCR-activated Erk–AMPK signaling pathway. IEE Proc Syst Biol 153: 201–211.Y. ZhengA. Rundell2006Comparative study of parameter sensitivity analyses of the TCR-activated Erk–AMPK signaling pathway.IEE Proc Syst Biol153201211
  36. 36. Maurya MR, Bornheimer SJ, Venkatasubramanian V, Subramaniam S (2005) Reduced-order modeling of biochemical networks: Application to the GTPase-cycle signaling module. IEE Proc Syst Biol 152: 229–242.MR MauryaSJ BornheimerV. VenkatasubramanianS. Subramaniam2005Reduced-order modeling of biochemical networks: Application to the GTPase-cycle signaling module.IEE Proc Syst Biol152229242
  37. 37. McKay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 2: 239–245.MD McKayRJ BeckmanWJ Conover1979A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.Technometrics2239245
  38. 38. Helton JC, Davis FJ (2000) Sampling-based methods. In: Satelli A, Chan K, Scott EM, editors. Sensitivity analysis. New York: Wiley. pp. 101–153.JC HeltonFJ Davis2000Sampling-based methods.In:. A. SatelliK. ChanEM ScottSensitivity analysisNew YorkWiley101153
  39. 39. Downing DJ, Gardner RH, Hoffman FO (1985) An examination of response–surface methodologies for uncertainty analysis in assessment of models. Technometrics 27: 151–163.DJ DowningRH GardnerFO Hoffman1985An examination of response–surface methodologies for uncertainty analysis in assessment of models.Technometrics27151163
  40. 40. Kenakin T (1995) Pharmacological proteus? Trends Pharmacol Sci 16: 256–258.T. Kenakin1995Pharmacological proteus?Trends Pharmacol Sci16256258
  41. 41. Chidiac P, Nouet S, Bouvier M (1996) Agonist-induced modulation of inverse agonist efficacy at the beta 2-adrenergic receptor. Mol Pharmacol 50: 662–669.P. ChidiacS. NouetM. Bouvier1996Agonist-induced modulation of inverse agonist efficacy at the beta 2-adrenergic receptor.Mol Pharmacol50662669
  42. 42. Jansson CC, Kukkonen JP, Nasman J, Huifang GE, Wurster S, et al. (1998) Protean agonism at α2a-adrenoceptors. Mol Pharmacol 53: 963–968.CC JanssonJP KukkonenJ. NasmanGE HuifangS. Wurster1998Protean agonism at α2a-adrenoceptors.Mol Pharmacol53963968
  43. 43. Gbahou F, Rouleau A, Morisset S, Parmentier R, Crochet S, et al. (2003) Protean agonism at histamine H3 receptors in vitro and in vivo. Proc Natl Acad Sci U S A 100: 11086–11091.F. GbahouA. RouleauS. MorissetR. ParmentierS. Crochet2003Protean agonism at histamine H3 receptors in vitro and in vivo.Proc Natl Acad Sci U S A1001108611091
  44. 44. Pineyro G, Azzi M, deLean A, Schiller PW, Bouvier M (2005) Reciprocal regulation of agonist and inverse agonist signaling efficacy upon short-term treatment of the human delta–opioid receptor with an inverse agonist. Mol Pharmacol 67: 336–348.G. PineyroM. AzziA. deLeanPW SchillerM. Bouvier2005Reciprocal regulation of agonist and inverse agonist signaling efficacy upon short-term treatment of the human delta–opioid receptor with an inverse agonist.Mol Pharmacol67336348
  45. 45. Jansson CC, Savola JM, Akerman KEO (1994) Different sensitivity of α2A-C10 and α2A-C4 receptor subtypes in coupling to inhibition of cAMP accumulation. Biochem Biophys Res Commun 199: 869–875.CC JanssonJM SavolaKEO Akerman1994Different sensitivity of α2A-C10 and α2A-C4 receptor subtypes in coupling to inhibition of cAMP accumulation.Biochem Biophys Res Commun199869875
  46. 46. Jansson CC, Marjamaki A, Luomala K, Savola JM, Scheinin M, et al. (1994) Coupling of human α2-adrenoceptor subtypes to regulation of cAMP production in transfected S115 cells. Eur J Pharmacol 266: 165–174.CC JanssonA. MarjamakiK. LuomalaJM SavolaM. Scheinin1994Coupling of human α2-adrenoceptor subtypes to regulation of cAMP production in transfected S115 cells.Eur J Pharmacol266165174
  47. 47. Yi T-M, Kitano H, Simon MI (2003) A quantitative characterization of the yeast heterotrimeric G protein cycle. Proc Natl Acad Sci U S A 100: 10764–10769.T-M YiH. KitanoMI Simon2003A quantitative characterization of the yeast heterotrimeric G protein cycle.Proc Natl Acad Sci U S A1001076410769
  48. 48. Hoffmann C, Gaietta G, Bunemann M, Adams SR, Oberdorff-Maass S, et al. (2005) A FlAsH-based FRET approach to determine G protein–coupled receptor activation in living cells. Nat Methods 2: 171–176.C. HoffmannG. GaiettaM. BunemannSR AdamsS. Oberdorff-Maass2005A FlAsH-based FRET approach to determine G protein–coupled receptor activation in living cells.Nat Methods2171176
  49. 49. Gales C, Rebois RV, Hogue M, Trieu P, Breit A, et al. (2005) Real-time monitoring of receptor and G protein interactions in living cells. Nat Methods 2: 177–184.C. GalesRV ReboisM. HogueP. TrieuA. Breit2005Real-time monitoring of receptor and G protein interactions in living cells.Nat Methods2177184
  50. 50. Kenakin T (2001) Inverse, protean, and ligand-selective agonism: Matters of receptor conformation. FASEB J 15: 598–611.T. Kenakin2001Inverse, protean, and ligand-selective agonism: Matters of receptor conformation.FASEB J15598611
  51. 51. Kenakin T (1997) Differences between natural and recombinant G protein–coupled receptor systems with varying receptor/G protein stoichiometry. Trends Pharmacol Sci 18: 456–464.T. Kenakin1997Differences between natural and recombinant G protein–coupled receptor systems with varying receptor/G protein stoichiometry.Trends Pharmacol Sci18456464
  52. 52. Seifert R, Lee TW, Lam VT, Kobilka BK (1998) Reconstitution of beta2–adrenoceptor–GTP-binding-protein interaction in Sf9 cells—High coupling efficiency in a beta2–adrenoceptor–G(s alpha) fusion protein. Eur J Biochem 255: 369–382.R. SeifertTW LeeVT LamBK Kobilka1998Reconstitution of beta2–adrenoceptor–GTP-binding-protein interaction in Sf9 cells—High coupling efficiency in a beta2–adrenoceptor–G(s alpha) fusion protein.Eur J Biochem255369382
  53. 53. Ping P, Hammond HK (1994) Diverse G protein and beta-adrenergic receptor mRNA expression in normal and failing porcine hearts. Am J Physiol 267: 20779–20785.P. PingHK Hammond1994Diverse G protein and beta-adrenergic receptor mRNA expression in normal and failing porcine hearts.Am J Physiol2672077920785
  54. 54. Leopoldt D, Harteneck C, Nurnberg B (1997) G proteins endogenously expressed in Sf9 cells: Interactions with mammalian histamine receptors. Naunyn-Schmiedeberg's Arch Pharmacol 356: 216–224.D. LeopoldtC. HarteneckB. Nurnberg1997G proteins endogenously expressed in Sf9 cells: Interactions with mammalian histamine receptors.Naunyn-Schmiedeberg's Arch Pharmacol356216224
  55. 55. Remmers AE, Engel C, Liu M, Neubig RR (1999) Interdomain interactions regulate GDP release from heterotrimeric G proteins. Biochemistry 38: 13795–13800.AE RemmersC. EngelM. LiuRR Neubig1999Interdomain interactions regulate GDP release from heterotrimeric G proteins.Biochemistry381379513800
  56. 56. McKernan RM, Strickland WR, Insel PA (1988) Selective blockade and recovery of cell surface alpha 2–adrenergic receptors in human erythroleukemia (HEL) cells. Studies with the irreversible antagonist benextramine. Mol Pharmacol 33: 51–57.RM McKernanWR StricklandPA Insel1988Selective blockade and recovery of cell surface alpha 2–adrenergic receptors in human erythroleukemia (HEL) cells. Studies with the irreversible antagonist benextramine.Mol Pharmacol335157
  57. 57. Cotecchia S, Kobilka BK, Daniel KW, Nolan RD, Lapetina EY, et al. (1990) Multiple second messenger pathways of alpha-adrenergic receptor subtypes expressed in eukaryotic cells. J Biol Chem 265: 63–69.S. CotecchiaBK KobilkaKW DanielRD NolanEY Lapetina1990Multiple second messenger pathways of alpha-adrenergic receptor subtypes expressed in eukaryotic cells.J Biol Chem2656369
  58. 58. Lan KL, Zhong H, Nanamori M, Neubig RR (2000) Rapid kinetics of regulator of G protein signaling (RGS)–mediated Galphai and Galphao deactivation. Galpha specificity of RGS4 and RGS7. J Biol Chem 275: 33497–33503.KL LanH. ZhongM. NanamoriRR Neubig2000Rapid kinetics of regulator of G protein signaling (RGS)–mediated Galphai and Galphao deactivation. Galpha specificity of RGS4 and RGS7.J Biol Chem2753349733503
  59. 59. Kleymann G, Boege F, Hahn M, Hampe W, Vasudevan S, et al. (1993) Human beta 2–adrenergic receptor produced in stably transformed insect cells is functionally coupled via endogenous GTP-binding protein to adenylyl cyclase. Eur J Biochem 213: 797–804.G. KleymannF. BoegeM. HahnW. HampeS. Vasudevan1993Human beta 2–adrenergic receptor produced in stably transformed insect cells is functionally coupled via endogenous GTP-binding protein to adenylyl cyclase.Eur J Biochem213797804
  60. 60. Ransnas LA, Svoboda P, Jasper JR, Insel PA (1989) Stimulation of beta-adrenergic receptors of S49 lymphoma cells redistributes the alpha subunit of the stimulatory G protein between cytosol and membranes. Proc Natl Acad Sci U S A 86: 7900–7903.LA RansnasP. SvobodaJR JasperPA Insel1989Stimulation of beta-adrenergic receptors of S49 lymphoma cells redistributes the alpha subunit of the stimulatory G protein between cytosol and membranes.Proc Natl Acad Sci U S A8679007903
  61. 61. Dohlman HG, Thorner J (1997) RGS proteins and signaling by heterotrimeric G proteins. J Biol Chem 272: 3871–3874.HG DohlmanJ. Thorner1997RGS proteins and signaling by heterotrimeric G proteins.J Biol Chem27238713874
  62. 62. Lan KL, Sarvazyan NA, Taussig R, Mackenzie RG, DiBello PR, et al. (1998) A point mutation in G alpha(o) and G alpha(i1) blocks interaction with regulator of G protein signaling proteins. J Biol Chem 273: 12794–12797.KL LanNA SarvazyanR. TaussigRG MackenziePR DiBello1998A point mutation in G alpha(o) and G alpha(i1) blocks interaction with regulator of G protein signaling proteins.J Biol Chem2731279412797
  63. 63. Clark MJ, Harrison C, Zhong H, Neubig RR, Traynor JR (2003) Endogenous RGSs protein action modulates {micro}–opioid signaling through Galpha o: Effects on adenylyl cyclase, extracellular signal-regulated kinases, and intracellular calcium pathways. J Biol Chem 278: 9418–9425.MJ ClarkC. HarrisonH. ZhongRR NeubigJR Traynor2003Endogenous RGSs protein action modulates {micro}–opioid signaling through Galpha o: Effects on adenylyl cyclase, extracellular signal-regulated kinases, and intracellular calcium pathways.J Biol Chem27894189425
  64. 64. Jin Y, Zhong H, Omnaas JR, Neubig RR, Mosberg HI (2004) Structure-based design, synthesis, and activity of peptide inhibitors of RGS4 GAP activity. Methods Enzymol 389: 266–277.Y. JinH. ZhongJR OmnaasRR NeubigHI Mosberg2004Structure-based design, synthesis, and activity of peptide inhibitors of RGS4 GAP activity.Methods Enzymol389266277
  65. 65. Grant KR, Harnett W, Milligan G, Harnett MM (1997) Differential G protein expression during B- and T-cell development. Immunology 90: 564–571.KR GrantW. HarnettG. MilliganMM Harnett1997Differential G protein expression during B- and T-cell development.Immunology90564571
  66. 66. Hadcock JR, Malbon CC (1993) Agonist regulation of gene expression of adrenergic receptors and G proteins. J Neurochem 60: 1–9.JR HadcockCC Malbon1993Agonist regulation of gene expression of adrenergic receptors and G proteins.J Neurochem6019
  67. 67. Ping P, Gelzer BR, Roth DA, Kiel D, Insel PA, et al. (1995) Reduced beta-adrenergic receptor activation decreases G protein expression and beta-adrenergic receptor kinase activity in porcine heart. J Clin Invest 95: 1271–1280.P. PingBR GelzerDA RothD. KielPA Insel1995Reduced beta-adrenergic receptor activation decreases G protein expression and beta-adrenergic receptor kinase activity in porcine heart.J Clin Invest9512711280
  68. 68. Feldman AM, Cates AE, Veazey WB, Hershberger RE, Bristow MR, et al. (1988) Increase of the 40,000-mol wt pertussis toxin substrate (G protein) in the failing human heart. J Clin Invest 82: 189–197.AM FeldmanAE CatesWB VeazeyRE HershbergerMR Bristow1988Increase of the 40,000-mol wt pertussis toxin substrate (G protein) in the failing human heart.J Clin Invest82189197
  69. 69. Eschenhagen T, Mende U, Nose M, Schmitz M, Scholz H, et al. (1992) Increased messenger RNA level of the inhibitory G protein alpha subunit Gi alpha-2 in human end-stage heart failure. Circ Res. 70. T. EschenhagenU. MendeM. NoseM. SchmitzH. Scholz1992Increased messenger RNA level of the inhibitory G protein alpha subunit Gi alpha-2 in human end-stage heart failure.Circ Res70
  70. 70. Post SR, Hammond HK, Insel PA (1999) Beta-adrenergic receptors and receptor signaling in heart failure. Annu Rev Pharmacol Toxicol 39: 343–360.SR PostHK HammondPA Insel1999Beta-adrenergic receptors and receptor signaling in heart failure.Annu Rev Pharmacol Toxicol39343360
  71. 71. Chakir K, Xiang Y, Yang DM, Zhang SJ, Cheng HP, et al. (2003) The third intracellular loop and the carboxyl terminus of beta(2)-adrenergic receptor confer spontaneous activity of the receptor. Mol Pharmacol 64: 1048–1058.K. ChakirY. XiangDM YangSJ ZhangHP Cheng2003The third intracellular loop and the carboxyl terminus of beta(2)-adrenergic receptor confer spontaneous activity of the receptor.Mol Pharmacol6410481058
  72. 72. Ghanouni P, Steenhuis JJ, Farrens DL, Kobilka BK (2001) Agonist-induced conformational changes in the G protein–coupling domain of the beta 2 adrenergic receptor. Proc Natl Acad Sci U S A 98: 5997–6002.P. GhanouniJJ SteenhuisDL FarrensBK Kobilka2001Agonist-induced conformational changes in the G protein–coupling domain of the beta 2 adrenergic receptor.Proc Natl Acad Sci U S A9859976002
  73. 73. Peleg G, Ghanouni P, Kobilka BK, Zare RN (2001) Single-molecule spectroscopy of the beta 2 adrenergic receptor: Observation of conformational substates in a membrane protein. Proc Natl Acad Sci U S A 98: 8469–8474.G. PelegP. GhanouniBK KobilkaRN Zare2001Single-molecule spectroscopy of the beta 2 adrenergic receptor: Observation of conformational substates in a membrane protein.Proc Natl Acad Sci U S A9884698474
  74. 74. Gether U, Lin S, Kobilka B (1995) Fluorescent labeling of purified β2 adrenergic receptor: Evidence for ligand-specific conformational changes. J Biol Chem 270: 28268–28275.U. GetherS. LinB. Kobilka1995Fluorescent labeling of purified β2 adrenergic receptor: Evidence for ligand-specific conformational changes.J Biol Chem2702826828275
  75. 75. Cohen BE, Pralle A, Yao XJ, Swaminath G, Gandhi CS, et al. (2005) A fluorescent probe designed for studying protein conformational change. Proc Natl Acad Sci U S A 102: 965–970.BE CohenA. PralleXJ YaoG. SwaminathCS Gandhi2005A fluorescent probe designed for studying protein conformational change.Proc Natl Acad Sci U S A102965970
  76. 76. Lohse MJ, Engelhardt S, Eschenhagen T (2003) What is the role of beta-adrenergic signaling in heart failure? Circ Res 93: 896–906.MJ LohseS. EngelhardtT. Eschenhagen2003What is the role of beta-adrenergic signaling in heart failure?Circ Res93896906
  77. 77. Pavia J, de Ceballos ML, Sanchez de la Cuesta F (1998) Alzheimer's disease: Relationship between muscarinic cholinergic receptors, beta-amyloid and tau proteins. Fundam Clin Pharmacol 12: 473–481.J. PaviaML de CeballosF. Sanchez de la Cuesta1998Alzheimer's disease: Relationship between muscarinic cholinergic receptors, beta-amyloid and tau proteins.Fundam Clin Pharmacol12473481
  78. 78. El-Samad H, Kurata H, Doyle JC, Gross CA, Khammash M (2005) Surviving heat shock: Control strategies for robustness and performance. Proc Natl Acad Sci U S A 102: 2736–2741.H. El-SamadH. KurataJC DoyleCA GrossM. Khammash2005Surviving heat shock: Control strategies for robustness and performance.Proc Natl Acad Sci U S A10227362741
  79. 79. Yi T-M, Huang Y, Simon MI, Doyle J (2000) Robust perfect adaptation in bacterial chemotaxis through integral feedback control. Proc Natl Acad Sci U S A 97: 4649–4653.T-M YiY. HuangMI SimonJ. Doyle2000Robust perfect adaptation in bacterial chemotaxis through integral feedback control.Proc Natl Acad Sci U S A9746494653
  80. 80. Bagci E, Vodovotz Y, Billiar T, Ermentrout G, Bahar I (2006) Bistability in apoptosis: Roles of Bax, Bcl-2, and mitochondrial permeability transition pores. Biophys J 90: 1546–1559.E. BagciY. VodovotzT. BilliarG. ErmentroutI. Bahar2006Bistability in apoptosis: Roles of Bax, Bcl-2, and mitochondrial permeability transition pores.Biophys J9015461559
  81. 81. Linderman JJ (2000) Kinetic approaches to understanding ligand efficacy. In: Kenakin T, Angus J, editors. The pharmacology of functional, biochemical, and recombinant receptor systems. Handbook of experimental pharmacology. New York: Springer-Verlag. pp. 119–146.JJ Linderman2000Kinetic approaches to understanding ligand efficacy.In:. T. KenakinJ. AngusThe pharmacology of functional, biochemical, and recombinant receptor systems. Handbook of experimental pharmacologyNew YorkSpringer-Verlag119146
  82. 82. Vilar JMG, Jansen R, Sander C (2006) Signal processing in the TGF-beta; superfamily ligand–receptor network. PLoS Comp Biol 2(1): e3.. JMG VilarR. JansenC. Sander2006Signal processing in the TGF-beta; superfamily ligand–receptor network.PLoS Comp Biol21e3.
  83. 83. Wiley HS, Shvartsman SY, Lauffenburger DA (2003) Computational modeling of the EGF–receptor system: A paradigm for systems biology. Trends Cell Biol 13: 43–50.HS WileySY ShvartsmanDA Lauffenburger2003Computational modeling of the EGF–receptor system: A paradigm for systems biology.Trends Cell Biol134350
  84. 84. Campagne F, Neves S, Chang CW, Skrabanek L, Ram PT, et al. (2004) Quantitative information management for the biochemical computation of cellular networks. Science STKE 248: I11.F. CampagneS. NevesCW ChangL. SkrabanekPT Ram2004Quantitative information management for the biochemical computation of cellular networks.Science STKE248I11
  85. 85. Gilman AG, Simon MI, Bourne HR, Harris BA, Long R, et al. (2002) Overview of the alliance for cellular signaling. Nature 420: 703–706.AG GilmanMI SimonHR BourneBA HarrisR. Long2002Overview of the alliance for cellular signaling.Nature420703706
  86. 86. Mendes P, Bulmore DL, Farmer AD, Steadman PA, Waugh ME, et al. (2000) PathDB: A second generation metabolic database. In: Hofmeyr JHS, Rohwer JM, Snoep JL, editors. Animating the cellular map. Stellenbosch: Stellenbosch University Press. pp. 207–212.P. MendesDL BulmoreAD FarmerPA SteadmanME Waugh2000PathDB: A second generation metabolic database.In:. JHS HofmeyrJM RohwerJL SnoepAnimating the cellular mapStellenboschStellenbosch University Press207212
  87. 87. Blower SM, Dowlatabadi H (1994) Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example. Intl Stat Rev 62: 229–243.SM BlowerH. Dowlatabadi1994Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example.Intl Stat Rev62229243
  88. 88. Wyss GD, Jorgensen KH (1998) A user's guide to LHS—Sandia's Latin Hypercube Sampling software. SAND98–0210. Sandia (New Mexico): Sandia National Laboratory. GD WyssKH Jorgensen1998A user's guide to LHS—Sandia's Latin Hypercube Sampling software. SAND98–0210Sandia (New Mexico)Sandia National Laboratory
  89. 89. Kenakin T (1997) Molecular pharmacology: A short course. Cambridge (Massachusetts): Blackwell Science. 235 p.T. Kenakin1997Molecular pharmacology: A short courseCambridge (Massachusetts)Blackwell Science235
  90. 90. Kenakin T (1996) The classification of seven transmembrane receptors in recombinant expression systems. Pharmacol Rev 48: 413–463.T. Kenakin1996The classification of seven transmembrane receptors in recombinant expression systems.Pharmacol Rev48413463
  91. 91. Shea LD, Omann GM, Linderman JJ (1997) Calculation of diffusion-limited kinetics for the reactions in collision coupling and receptor cross-linking. Biophys J 73: 2949–2959.LD SheaGM OmannJJ Linderman1997Calculation of diffusion-limited kinetics for the reactions in collision coupling and receptor cross-linking.Biophys J7329492959
  92. 92. Neubig RR (1994) Membrane organization in G protein mechanisms. FASEB J 8: 939–946.RR Neubig1994Membrane organization in G protein mechanisms.FASEB J8939946
  93. 93. Sklar LA (1987) Real-time spectroscopic analysis of ligand–receptor dynamics. Annu Rev Biophys Biophys Chem 16: 479–506.LA Sklar1987Real-time spectroscopic analysis of ligand–receptor dynamics.Annu Rev Biophys Biophys Chem16479506
  94. 94. Ransnas LA, Insel PA (1988) Quantitation of the guanine nucleotide binding regulatory protein Gs in S49 cell membranes using antipeptide antibodies to alpha s. J Biol Chem 263: 9482–9485.LA RansnasPA Insel1988Quantitation of the guanine nucleotide binding regulatory protein Gs in S49 cell membranes using antipeptide antibodies to alpha s.J Biol Chem26394829485
  95. 95. Alt A, McFadyen IJ, Fan CD, Woods JH, Traynor JR (2001) Stimulation of guanosine-5′-o-(3-[35s]thio)triphosphate binding in digitonin-permeabilized C6 rat glioma cells: Evidence for an organized association of mu-opioid receptors and G protein. J Pharmacol Exp Ther 298: 116–121.A. AltIJ McFadyenCD FanJH WoodsJR Traynor2001Stimulation of guanosine-5′-o-(3-[35s]thio)triphosphate binding in digitonin-permeabilized C6 rat glioma cells: Evidence for an organized association of mu-opioid receptors and G protein.J Pharmacol Exp Ther298116121