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MOLECULAR PHARMACOLOGY 52:144-154 (1997).
The Stability of the Agonist 
2-Adrenergic
Receptor-Gs Complex: Evidence for Agonist-Specific States
Andrejs M.
Krumins and
Roger
Barber
Department of Integrative Biology, Pharmacology, and
Physiology, University of Texas-Houston Medical School, Houston,
Texas 77225-0334
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Summary |
A restricted version of the ternary complex model for receptor-G
protein complex formation has recently been proposed. Known as the
two-state model, this model proposes that in the context of agonist and
G protein interactions, only two thermodynamic states exist for the
receptor: active (R*) and inactive (R). One form of this model suggests
that only the R* state of the receptor is capable of interacting with
and subsequently activating G proteins. We directly tested the kinetic
aspects of a strict two-state receptor model in a cell line containing
the native 
2-adrenergic receptor that is capable of
inducing Gs expression. We examined adenylyl cyclase
activity in the presence of limiting GTP levels and concluded that
there exists a different rate of heterotrimer dissociation (i.e., HR*G
yields HR* + G*) for different 2-agonists. This finding is inconsistent with a strict two-state model in which R* is a characteristic of the receptor that is independent of the identity of
the agonist. It implies that agonist activation of adenylyl cyclase is
more complicated than a simple two-state model.
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Introduction |
The concept of agonist-induced
receptor conformation has been the subject of much debate. Although the
order of the interaction between agonists, receptors, and G proteins is
relatively undisputed, the stoichiometry and nature of the proteins
during activation remain very much in question. One model in particular
that seems to be gathering widespread acceptance for G protein-coupled
receptors is the two-state model for receptor activation (1-3). The
two-state model is a restricted version of the ternary complex model
(4) and has been extremely useful for explaining the experimental observations involving 2-adrenergic receptor activation
of adenylyl cyclase. Moreover, its stark simplicity makes it easy to
set up arguments that attempt to predict the behavior of other features of the system.
In physicochemical terms, the two-state model cannot be exactly true.
Molecules exist in a very large number of conformational states, and
trying to describe the thermodynamic properties of a receptor with just
two conformational states must be an approximation. That given, the
usefulness of the model depends on just how reliable its predictions
are. In terms of receptor-G protein interactions, a strict
two-state model is as shown here:
where G is the G protein, R is receptor, and H is agonist. It is
supposed that the receptor must exist in one of two conformations (R
and R*), with both H and G binding with greater affinity to the R*
form. Given this condition, there automatically is cooperativity in
binding of the G and H to the R (which explains the GTP shift). Moreover, the different efficiencies of different agonists to promote
adenylyl cyclase activation is easily explained by supposing that
stronger agonists favor binding to the R* form to a greater extent than
do weaker agonists and hence more readily form the HR*G ternary
complex. In addition, inverse agonists tend to stabilize the inactive R
state (5, 6). It must be emphasized that in a strict two-state
model, the affinity of G for R is identical to that for HR and its
affinity for R* is identical to that for HR* because R and R*
represent unique defined conformational states (or collection of
states). Given the above set-up, it is possible to develop the
thermodynamics of the system in terms of the relative concentrations of
the two conformations: when free and when bound to G protein, to
agonist or to both.
The classic development of the thermodynamics of the H, R, and G
interactions in terms of the ternary complex model (4) shown below is
not a one-state but rather a multistate approach. The use of a
different dissociation constants for the binding of G to HR than for
the binding to R is an acknowledgment that the receptor in HR is in a
different conformational state than in R
In this equation, therefore, unlike that for the two-state model,
R, HR, and so on represent the totality of possible conformations and
not a single conformation or a defined group of conformations in a
constant proportion. Thus, although it is superficially simpler than
Eq. 1, Eq. 2 is thermodynamically complete and, unlike Eq. 1, it is
guaranteed to be correct if the ternary complex model is correct as
drawn. Eq. 2 may be compared to a true one-state model shown below in
Eq. 3. In this equation, because the receptor can exist in only one
conformation, there can be no cooperativity of binding between G
protein and agonist.
The scheme proposed by Samama et al. (1) (shown below
in Eq. 4), although described in that publication as a two-state scheme, is in fact a multistate model. The fact that different affinities are used for the binding of H to the R* and the R*G state
means that a different conformation is assumed for R* when bound and
when not bound to G protein. In other words, there are at least three
states. The use of this scheme enables one to be completely accurate
with the thermodynamics, but it prevents one from making the
simplistic twostate arguments, which are the attraction of the
model.
For the two-state model to be useful in successfully making
predictions not possible by the classic thermodynamic approach, additional assumptions beyond the schemes discussed above are necessary. If the assumptions are limited to proposing that agonists bind more strongly to the conformation that also more strongly binds G
protein, then no argument can be drawn beyond those already possible
for the classic thermodynamic scheme. Similarly, negative agonists that
reduce the amount of adenylyl cyclase activity below the basal level
found in the absence of ligand can be equally well explained (and
predicted) by the classic thermodynamic approach. Where the strict
two-state model potentially offers advantages is in quantification of
predictions and as a potential for a quantitative relationship between
thermodynamic properties (e.g., GTP shifts) and kinetic features (e.g.,
efficiencies of agonists in adenylyl cyclase activation).
In the current study, we investigated the specific hypothesis that the
fraction of ternary complex in the activating conformation (i.e., in
the HR*G form) determines the rate at which GDP/GTP exchange occurs. In
the strict two-state model, the HR*G conformation is the same for all
agonists, which differ only in the ratio of HR*G to
HRGtotal [where HRGtotal = HRG (inactive R) + HR*G (active R)]. The rationale for the test of this hypothesis is to
determine the rate of breakdown of the ternary complex for four
different 2 agonists. It will be shown that in all the
cases the thermodynamics requires that >85% of the ternary complex
must be in the HR*G rather than the HRG form. Proportionality of the
rate of breakdown for the ternary complex with the fraction of ternary
complex that is HR*G would therefore require that all the rates be
similar. This did not occur. We will therefore conclude that the
two-state model in its strictest form cannot be applied in all cases.
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Experimental Procedures |
Materials.
Molecular biology reagents, Dulbecco's modified
Eagle's medium, and geneticin were obtained from GIBCO BRL
(Gaithersburg, MD). Tris Base, GTP, and
guanosine-5
-O-(3-thio)triphosphate were from
Boehringer-Mannheim Biochemicals (Indianapolis, IN).
[
-32P]ATP, Na-125I, and
[2,8-3H]cAMP were from DuPont-New England Nuclear
(Boston, MA). Dexamethasone and the remaining reagents from Sigma
Chemical (St. Louis, MO).
Cell culture of an inducible Gs
cell line.
The establishment and characterization of a stably transfected murine
S49 cyc
T cell lymphoma capable of inducing
Gs protein after dexamethasone treatment have been
previously described (7). Briefly, the S49 cyc cell line,
lacking Gs mRNA and protein (8), was electroporated with
the 7.7-kb pMMTV · Gs neo vector (a generous gift from J. Gonzales; described in detail in Ref. 9). The vector contains the
cDNA encoding rat Gslong linked downstream of the
dexamethasone-inducible mouse mammary tumor virus long terminal repeat
promoter. The vector also contains the selection marker for neomycin
resistance, neomycin phosphotransferase, constitutively driven by the
human -globin promoter.
Transfected drug-resistant cell lines (S49*cyc) were
maintained in stock tissue culture flasks (Corning Glassworks, Corning, NY) at 37° in HEPES-buffered Dulbecco's modified Eagle's medium supplemented with penicillin, streptomycin, 10% heat-treated horse serum, and 200 µg/ml geneticin to maintain selective pressure. Cells
were expanded for Gs gene induction experiments into eight individual preconditioned 2-liter roller bottles (Corning) from a
single stock source. The final cell density was 1 × 106 cells/ml when the volume was made up to 2 liters with
fresh Dulbecco's modified Eagle's medium plus 10% horse serum media.
Gs protein induction was initiated when the
S49*cyc cells were incubated with 5 µM
dexamethasone (final, added from a 10 mg/ml stock prepared in 95%
ethanol) for times of 1-24 hr. Transfected control cells underwent no
treatment.
Membrane preparations.
Cell membranes were prepared and
isolated as follows. Cells were washed twice with an excess of buffer A
(137 mM NaCl, 5.36 mM KCl, 1.1 mM
KH2PO4, and 1.08 mM
Na2HPO4, pH 7.2) by centrifugation at 600 × g. The cells were then resuspended in ice-cold cell lysis buffer B (20 mM HEPES, 150 mM NaCl, 5 mM NaH2PO4, 1 mM EDTA,
and 1 mM benzamidine, pH 7.4; buffer B also contained 10 µg/ml trypsin inhibitor and 10 µg/ml leupeptin to protect
Gs from possible proteolysis) and placed in a Parr bomb
(at 500 p.s.i.) for 25 min. The disrupted cells were centrifuged
for 5 min at 600 × g to pellet nuclear debris. The supernatant
was layered onto a 23% and 43% sucrose step gradient in HE buffer (20 mM HEPES and 1 mM EDTA, pH 8.0) and centrifuged
at 25,000 rpm in a Beckman Instruments (Columbia, MD) SW 28 rotor for
45 min at 4°. The membrane fraction was collected as a band at the
sucrose interface. The membranes were immediately frozen in liquid
N2 and stored at 
80°. Membrane concentrations were
determined using the BioRad (Hercules, CA) assay (10).
2 Agonist competition binding: GTP shift
analysis.
GTP shift binding analyses were used to analyze the
total amount of ternary complex (HRGtotal) present in the
S49*cyc membrane preparations containing different
Gs levels as [GTP] 
0. Binding analyses were carried
out in 500-µl reactions with a single 80 pM concentration
of the radiolabeled 2 antagonist 125I-CYP
[prepared according to the protocol of Barovsky et al. (11) with modifications by Hoyer et al. (12)] in the presence of increasing 2 agonist concentrations and the following
final concentrations of the reagents: 1 mM EDTA, pH 7.4, 20 mM HEPES, pH 7.4, 10 µM phentolamine, 0.3 mM MgCl2, and 20-50 µg of cell membranes
(diluted with HE, pH 8.0). The binding reaction was conducted in the
presence and absence of 10 µM
guanosine-5-O-(3-thio)triphosphate for 55 min at 30°. In
some experiments, binding was conducted under adenylyl cyclase assay
conditions (see below) to examine the effect of varied GTP
concentrations on the area of the GTP shift. In all experiments, the
nonspecific 125I-CYP binding was determined in the presence
of 10 µM alprenolol. Reactions were terminated with the
addition of 2.5 ml of ice-cold stop buffer (50 mM Tris-Cl,
pH 7.4, 10 mM MgCl2) followed by the immediate
filtration of the solution [with the use of a Millipore (Bedford, MA)
1225 vacuum filtration apparatus] onto Whatman (Clifton, NJ) GF/C
filters. The reaction tubes were rinsed with an additional 2.5 ml of
ice-cold stop buffer. The GF/C filters were washed four or five times
with 2.5 ml of stop buffer, dried, and removed to scintillation vials,
in which 125I-CYP activity was counted on a Beckman Gamma
4000 System Counter for 1 min. Activity, measured as counts per minute,
was converted into units of fmol/mg with the aid of a spread-sheet
program using Lotus 1-2-3. The specific binding of 125I-CYP
for each concentration was determined as mean values of triplicate
measurements for total binding less the mean values of triplicate
measurements for nonspecific binding. Data were plotted as log
[agonist] versus the normalized fraction of agonist bound and were
subsequently analyzed using our recently developed Scatchard method
(15) to generate agonist affinity constants (to R and RG, respectively)
and [Gs]total and HRGtotal
levels. The use of two-component nonlinear regression analyses
(GraphPAD) to analyze the GTP curve for HRGtotal levels
resulted in values similar to those obtained according to the Scatchard
method (±2.5%). The area for the GTP shifts were determined by the
trapezoid method. Comparisons of data ± standard deviations were
performed using single-factor analysis of variance
(p < 0.05).
Adenylyl cyclase assays.
Adenylyl cyclase assays, conducted
with saturating agonist levels and decreasing GTP concentrations, were
used to compare the rate of the ternary complex breakdown for different
agonists. The premise for using these assays is described in greater
detail below and in Discussion. Briefly, it is known experimentally
that at normal (high) GTP concentration, the formation of the
heterotrimer complex (HRGtotal) is rate limiting, and once
formed, it interacts with GTP extremely rapidly (13). For a strict
two-state model, the rate at which the HR*G complex dissociates should
be independent of the agonist used to stabilize the R* conformation;
therefore, the rate of HRGtotal breakdown should be
proportional to the fraction of HR*G formed by each agonist. The rate
of HR*G dissociation was made rate limiting by reducing GTP and
allowing the HRGtotal complex to accumulate.
Adenylyl cyclase activity was examined in S49*cyc
membranes containing different Gs levels using the
conditions for adenylyl cyclase assays as described by Clark et
al. (14), with some modifications. Briefly, adenylyl cyclase
activity in plasma membranes (~0.2 µg/µl, final) was brought to
steady state for 3 min at 30° in a 1800-µl volume in a 15 × 100 mm borosilicate glass tube with the following reagents (given as
final concentrations): 40 mM HEPES, pH 7.7, 1 mM EDTA, 0.3 mM MgCl2, 8 mM creatine phosphate, 16 units/ml creatine phosphokinase,
0.05 mM ATP, and 0.1 mM
3-isobutyl-1-methylxanthine. The preincubation period was necessary to
bring the HRGtotal levels for each agonist to equilibrium.
[For any agonist to be effective, the t1/2 of
HRGtotal formation cannot be less than the
t1/2 for adenylyl cyclase inactivation
(t1/2 = 15 sec for k1
in S49 cells (16)]. Each assay tube consisted of a different GTP concentration (0, 10, 30, 100, and 300 nM, and 1, 3, and 10 µM), so that for a single agonist, a series of eight
independent time courses was required to produce a single GTP
dose-response curve. The final concentrations of agonist required to
saturate the 2-adrenergic receptors (i.e., occupy >90%
of receptor sites with agonist) were 10 µM for
epinephrine (Kd = 1100 nM), 2.2 µM for isoproterenol (Kd = 188 nM), 2.8 µM for fenoterol
(Kd = 400 nM), and 13 µM for dobutamine (Kd = 1400 nM).
Time course assays, spanning a 3-min period, were initiated with the
addition of 40 × 106 cpm [-32P]ATP
in a 200-µl aliquot containing the concentration of reagents described above. Time course assays were selected over a typical 10-min
adenylyl cyclase assay to examine the linearity of the response at low
GTP concentrations. The reactions were quenched every 18 sec by
removing 100-µl aliquots into 500 µl of ice-cold stop buffer. The
isolation and determination of [32P]cAMP activity were
performed as described by Salomon et al. (15). GTP-dependent
adenylyl cyclase dose-response curves were produced using linear
regression analysis (GraphPAD, San Diego, CA) to generate the slopes
(i.e., cyclase activity) from the linear cAMP accumulation versus time
plots. These data were subsequently transformed to Eadie-Hofstee plots
to examine the rate of HRGtotal breakdown (see below). All
comparisons of data were performed using single-factor analysis of
variance (p < 0.05, Excel).
Calculating the limiting HRGtotal and minimum HR*G
levels.
The limiting levels of HRGtotal, as [GTP]
0, were calculated from the Gs:R ratios obtained from
GTP shift analyses for each 2 agonist. The
Gs:R ratios, representing the maximum level of receptor-G
protein interaction in the presence of saturating agonist, were
obtained for each S49*cyc membrane preparation containing
variable Gs levels using a newly developed Scatchard method
(16, and data not shown); however, analysis of HR versus
HRGtotal plots for the GTP curves with a rectangular
hyperbola will yield similar results (±2.5%). With the
Gs:R level for an agonist, one can determine the limiting HRGtotal levels by using the following relationship (16):
</NU><DE>[HRG]<SUB>total</SUB></DE></FR>](/content/vol52/issue1/fulltext/144/img001.gif)
|
(5)
|
where KRG represents the dissociation
constant between HR and G. The values used for
KRG (unitless when [R]total is set
to unity and all concentrations are expressed as fractions or multiples of [R]total) were 0.007 ± 0.002 for epinephrine,
0.011 ± 0.006 for isoproterenol, 0.007 ± 0.003 for
fenoterol, and 0.029 ± 0.016 for dobutamine, and
[Gs]total:R ranged from 0.246 to 0.692 when epinephrine was used in GTP shift studies (16). Under saturating agonist conditions, [HR] 
[R]total, and the limiting
values for HRGtotal are determined by solving the following
quadratic equation:
![[HRG]<SUB>total</SUB><IT>=</IT>](/content/vol52/issue1/fulltext/144/img002.gif)
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(6)
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The minimum HR*G levels are calculated from the
HRGtotal levels using the argument developed in Appendix
II. The values for Kd1, representing the low
agonist affinity binding component, and Kd2,
representing the high agonist affinity binding component, were
generated using a two-component nonlinear regression curve analysis of
the GTP curve for each agonist (GraphPAD). The data representing the
limiting HRGtotal and minimum HR*G values for each agonist
were normalized with respect to the HRGtotal and HR*G
levels obtained for epinephrine in the same membrane preparations. The
data for different membrane preparations were averaged and analyzed
using single-factor analysis of variance (p < 0.05, Excel).
Determination of the rate of HRGtotal breakdown.
In a strict two-state model, the rate of HRGtotal breakdown
should be proportional to the fraction of HR*G formed by each agonist
(i.e., once HR*G is formed the rate of breakdown should be agonist
independent). The experimental rate of HRGtotal breakdown was determined by fitting the Eadie-Hofstee plots (with reversed axes
so that the x-axis represented 
and the y-axis
represented v/s) for cyclase activity to a
quartic solution that kinetically describes the GTP-dependent
activation of adenylyl cyclase (Appendix I). A formal derivation
reveals that the y-axis in the present case is
![<FR><NU>k<SUB>2</SUB>[HRG]<SUB>total</SUB>C<SUB>p</SUB></NU><DE><IT>k</IT><SUB>−<IT>1</IT></SUB></DE></FR>](/content/vol52/issue1/fulltext/144/img004.gif)
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(7)
|
where k1 is the rate of
G*s and G*sC inactivation,
[HRG]total is the limiting HRGtotal
concentrations, and Cp is the proportion of activatable
adenylyl cyclase that is dependent on the Gs level. As
[GTP] 0, [C] = [C]total, and is the same for all,
and the HR + G HRGtotal step is always fast
compared with HRGtotal breakdown; thus, the normal rate of
activation for various agonists does not matter. At the
y-axis, [GTP] = 0, and the intercept of the line with the
y-axis is given by:
![<FR><NU>k<SUB>2</SUB>[HRG]<SUB>total</SUB>C<SUB>p</SUB></NU><DE><IT>k</IT><SUB>−<IT>1</IT></SUB></DE></FR><IT>, </IT>where <FR><NU>[HRG]<SUB>total</SUB>C<SUB>p</SUB></NU><DE><IT>k</IT><SUB>−<IT>1</IT></SUB></DE></FR>](/content/vol52/issue1/fulltext/144/img005.gif)
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(8)
|
is the same for all agonists. Therefore, the intercepts on the
y-axis for the different agonists are in direct proportion to k2. The data representing
k2 values for each agonist were normalized with
respect to the k2 values obtained for
epinephrine in the same membrane preparations. As with the
HRGtotal and HR*G data, the differences in
k2 values were averaged and analyzed using a
single-factor analysis of variance (p < 0.05, Excel).
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Results |
Examination of adenylyl cyclase response with varying GTP
concentrations.
The kinetics of adenylyl cyclase activity, under
reduced GTP concentrations, were examined in S49*cyc
membranes containing variable Gs protein levels as
described in Experimental Procedures. At low GTP levels, the system
allows heterotrimeric (HRGtotal) complexes to accumulate,
which results in a change in the rate-limiting step of adenylyl cyclase
activation from the rate of heterotrimer formation to the rate of
active heterotrimer dissociation. The build-up of heterotrimer
complexes at low GTP concentrations was confirmed by conducting GTP
shift binding analyses with subsaturating GTP levels (4) using adenylyl cyclase assay conditions and demonstrating that the area of the GTP
shift was proportional to the level of GTP in the assay (data not
shown).
Fig. 1 compares the GTP dependence for adenylyl cyclase
activity in the presence of saturating concentrations of four different 2 agonists. In the absence of GTP, there was a small but
detectable adenylyl cyclase activity, which increased from 1.4 ± 0.4 pmol/min/mg for control membranes with minimal Gs
expression to 6.73 ± 2.2 pmol/min/mg at high Gs
levels. This increase in spontaneous activity presumably reflects the
increased Gs levels and the presence of endogenous guanine
nucleotides associated with the membrane preparation. Under saturating
agonist conditions, the presence of GTP resulted in increased response
until the response became saturated at ~3-10 µM GTP.
Linear regression analysis of the curves indicated that adenylyl
cyclase activity was always linear with time (r > 0.9), even under low GTP conditions, during the 3-min course of the assay.
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Fig. 1.
Comparison of adenylyl cyclase time course
studies in S49*cyc membranes stimulated with different
2-agonists. The four plots represent the adenylyl
cyclase responses to (A) epinephrine, (B) isoproterenol, (C) fenoterol,
and (D) dobutamine for a S49*cyc membrane preparation
containing the same level of Gs (Gs,
r = 0.7) in the presence of increasing GTP
concentrations ( , 0;  , 10 nM;  , 30 nM;
 , 100 nM;  , 300 nM;  , 1 µM;  , 3 µM;  , 10 µM).
cAMP accumulation was examined over a 3-min time period in the presence
of saturating concentrations of epinephrine (10 µM), isoproterenol (2.2 µM), fenoterol (2.8 µM),
and dobutamine (13 µM) as described in Experimental
Procedures. Linear regression analysis (GraphPAD) revealed values of
r > 0.95 for the curves. These data are representative
of the types of responses seen in S49*cyc membrane
preparations at seven significantly different Gs levels. The eight sets of membrane preparations constitute a series, and three
series were performed for epinephrine, whereas the data for
isoproterenol, fenoterol, and dobutamine represent measurements in a
single series.
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The magnitude of the adenylyl cyclase response was dependent on the
increasing Gs levels in the membrane and the type of
agonist used to stimulate activity. The rank order of efficacy
(epinephrine 
isoproterenol > fenoterol > dobutamine) can be seen clearly in a comparison of the normalized
adenylyl cyclase activity with respect to increasing GTP concentrations
(Fig. 2). Fig. 2 was generated by computing the slopes
of the individual linear responses shown in Fig. 1 using linear
regression analysis for each GTP concentration. Sigmoid nonlinear
regression curve analysis of the GTP response curves also revealed some
agonist dependency of the EC50 value for GTP, but in most
cases the EC50 value for GTP was ~100-150
nM. The data shown in Fig. 2 were transformed to
Eadie-Hofstee plots to analyze the rate of HRGtotal
breakdown (see below).
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Fig. 2.
Comparison of GTP dose response in
S49*cyc membranes with different
2-agonists. The GTP dose-response curves for epinephrine (), isoproterenol (), fenoterol ( ), and dobutamine () were generated by calculating the adenylyl cyclase activity from the slopes
of each curve in Fig. 1 using linear regression analysis (GraphPAD).
The cyclase activity ± standard deviation obtained from linear
regression analysis was normalized to a maximum epinephrine elicited
activity of 107.7 ± 1.5 pmol/min/mg. The data are demonstrated for a single Gs level ([Gs]total,
r = 0.7) and are representative of the differences
between agonist efficacy at seven other Gs levels.
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Determination of the limiting HRGtotal, HR*G, and rate
of HRG breakdown for each agonist.
The test of a true two-state
model for receptor activation lies in the assumption that the
experimental rate of HRGtotal dissociation should be
proportional to the fraction of activated HR*G complexes. This is
predicted to be so if the R* conformation is independent of the type of
agonist. Fig. 3 compares the relative levels of HRGtotal, the minimum HR*G, and rate of ternary complex
breakdown for different 2 agonists.
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Fig. 3.
Comparison of normalized HRG, HR*G, and rate of HRG
breakdown for different 2 agonists. The limiting HRG and
HR*G values were then calculated from the Gs:R ratios for
each agonist using the method described in Experimental Procedures. The
normalized values above represent the average ± standard error
from 24 assays for epinephrine, 8 assays for isoproterenol, assays for
fenoterol, and assays for dobutamine. With a value of 1.0 for
epinephrine, the normalized HRG values for isoproterenol, fenoterol,
and dobutamine are 0.92 ± 0.04, 0.92 ± 0.04, and 0.82 ± 0.09, respectively. The values for the minimum HR*G levels were
normalized with respect to epinephrine and are 0.906 ± 0.005, 0.915 ± 0.002, and 0.764 ± 0.006 for isoproterenol,
fenoterol, and dobutamine, respectively. The rates of breakdown for the
HRG complex for each agonist were determined by fitting the quartic
solution developed in Appendix I to adenylyl cyclase GTP dose-response
data as described in Experimental Procedures. The parameters for
[R]total and k1 were arbitrarily
set to 1.0. The values for k3 = 20 and
[C]total:[R]total = 0.45 were determined
previously using the methods of Krumins and Barber (16). The value for
the back-reaction constant k10 (0.012) was
selected with respect to the k1 value for
dobutamine (0.1) to be consistent with the level of HR*G depicted
above. The [Gs]total:[R]total
values in each S49*cyc membrane preparation were
dependent on the time of dexamethasone induction and were 0.271, control; 0.378, 1 hr; 0.492, 2 hr; 0.577, 4 hr; 0.659, 6 hr; 0.688, 8 hr; 0.705, 14 hr; and 0.665, 24 hr. At a constant Gs:R
level, the rate of breakdown, k2, served as the
only variable when agonist activity was compared. The rate of breakdown
data represents the average normalized differences ± standard
error between the agonists and epinephrine at each Gs
level. The data represent 24 experiments for epinephrine and eight
experiments for isoproterenol, fenoterol, and dobutamine.
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The limiting level of heterotrimer complex (HRGtotal)
formed in the absence of GTP was calculated for each agonist using GTP shift binding assays (Ref. 16 and data not shown) as described in
Experimental Procedures. With a value of 1.0 for epinephrine, the
normalized HRGtotal values (± standard error) for
isoproterenol, fenoterol, and dobutamine are 0.92 ± 0.04, 0.92 ± 0.04, and 0.82 ± 0.09, respectively. The
HRGtotal calculated levels for isoproterenol, fenoterol,
and dobutamine were consistently less than that for epinephrine
regardless of the Gs protein level in the plasma membrane (data not shown). An analysis of variance yielded no statistical difference for the effects of Gs protein levels on the
measured differences in HRGtotal levels between the
agonists and epinephrine. These data demonstrate that the
HRGtotal levels formed in the absence of GTP seem to be
relatively similar from the strong agonist epinephrine to the weak
agonist dobutamine.
The values for the minimum HR*G level for each of the agonists were
calculated as described in Appendix II and Experimental Procedures.
Fig. 3 demonstrates that the minimum fraction of active HR*G heterotrimer complexes makes up a significant proportion (>85%)
of the HRGtotal pool. The HR*G levels for each agonist were
normalized relative to the HR*G levels for epinephrine. The relative
minimum HR*G values (± standard error) for isoproterenol, fenoterol,
and dobutamine are 0.920 ± 0.001, 0.915 ± 0.002, and 0.764 ± 0.006, respectively. The 15% difference between HR*G and HRGtotal for dobutamine is thought to be due to the
difficulty in measuring Kd2, the high agonist
affinity component; it is highly dependent on the HRG 171 HR*G
distribution (in the HRGtotal pool) in that even a small
contribution of HRG (complex with inactive R) leads to a substantial
affect on the measured Kd2. The use of a
two-component nonlinear curve regression analysis to obtain a value for
the high affinity binding component (data not shown) revealed that the
difference between Kd1, the low affinity
component, and Kd2 averaged 28 ± 21-fold for dobutamine (16 experiments), 215 ± 70-fold for fenoterol (16 experiments), 232 ± 82-fold for isoproterenol (eight
experiments), and 400 ± 144-fold for epinephrine (24 experiments). The Kd2 values determined in this
fashion also seemed to decrease with increasing Gs levels
(r = 0.94), which may reflect a complicated receptor-G protein stoichiometry or the difficulty in determining the
high agonist affinity using a two-component curve-fitting analysis.
A comparison between the minimum HR*G levels and the relative rate of
ternary complex breakdown is also illustrated in Fig. 3. The rate of
ternary complex breakdown was determined from the y-intercept value of Eadie-Hofstee plots representing
GTP-dependent adenylyl cyclase activity for each agonist (see above).
The Eadie-Hofstee plots were used to fit a quartic solution describing
receptor-G protein/adenylyl cyclase interactions in the presence of
decreasing GTP (Appendix I). Using the conditions for fitting as described in Experimental Procedures, the x-axis represented
the number of active G*s-C complexes, a number that
is limited by C in this system (16). Therefore in practice, when
Gs 
C, the plots were linear; when Gs C,
the plot tails downward as it approaches the x-axis.
In our simulations of fitting the quartic solution to adenylyl cyclase
response data (data not shown), only a change in
k2 levels could result in different
y-intercepts for agonists of different efficacy (i.e.,
different k1 values). Using the quartic solution
to analyze experimental data revealed that the rates of breakdown
(normalized average ± standard error) for isoproterenol, fenoterol, and dobutamine were 0.906 ± 0.005, 0.738 ± 0.015, and 0.505 ± 0.011, respectively. No statistical
significance could be demonstrated for the effects of Gs
protein levels on the measured differences in the rates of heterotrimer
dissociation between the agonists and epinephrine. In addition, no
significant difference could be demonstrated between the minimum HR*G
levels and the rate of heterotrimer breakdown for either epinephrine or
isoproterenol. However, a significant difference could be demonstrated
for fenoterol (p = 4 × 103,
eight experiments) and dobutamine (p = 1.5 × 104; eight experiments). These data demonstrate that
the rate of breakdown is not similar for different 2
agonists (even when corrected for disparities in HR*G levels between
the agonists), implying that agonist activation of receptors is more
complicated than a simple two-state model.
| |
Discussion |
Two-state receptor activation model and the rate of
HRGtotal breakdown.
The extended ternary complex model
has provided an extremely useful thermodynamic description of
agonist/receptor-G protein interactions (1-3). This model proposes
that the receptor exists in one of two conformations: an inactive
state, R, which displays low agonist affinity and presumably does not
couple to G (2), and an active state, R*, which displays high agonist
affinity and couples to the G protein. The inactive state, R, can
isomerize into the active state R*, either spontaneously or with the
presence of an agonist. The agonist then stabilizes the R* form, and in general it is accepted that stronger agonists have a greater ability to
stabilize the active state.
The two-state model of receptor conformation offers advantages over the
more general classic ternary complex model only if it can make
verifiable predictions that the latter cannot. There are no qualitative
predictions that can be made by the two-state model that are not also
obvious from the classic model. The advantage, if there is one, is to
use the two-state system to make quantitative predictions based on the
amounts of the active form (R*, HR*, or HR*G) present in the system.
Essentially, this devolves to relating the quantities (or the relative
quantities) of these forms to effector activation or to the steady
state of effector activation under conditions in which the nature of
the agonist or antagonist is changed.
The scheme for the two-state ternary complex model as illustrated by
Samama et al. (1), in fact, depicts a multistate model in
which
is dependent on the factor . If there is complete validity to the
strict two-state model as described, = 1. Our data indicate 
1. The test for the two-state model of receptor conformation rested on
the following arguments:
1) We give measurements that show the actual accumulation of ternary
complex at [GTP] 0 for all the agonists used. In combination with
2) below, we can therefore calculate the maximum differences between
the agonists for the steady state concentrations of [HR*G].
2) If the two-state model is an adequate description of the
thermodynamics of ternary complex formation, then all of the agonists used are primarily in the HR*G state rather than in the HRG (R inactive) state in the ternary complex (HRGtotal).
3) We measure the adenylyl cyclase activity for all agonists as [GTP]
0. As [GTP] 0, the rate of breakdown of the ternary complex
is the rate-determining step in adenylyl cyclase activation. Measurement of adenylyl cyclase activity under these conditions therefore gives estimates for the relative rates of ternary complex conversion to active G*s.
4) The strict two-state model requires proportionality between the
amount of HR*G in the system at steady state and the rate of conversion
to active G*s.
It is known experimentally that at low values of [GTP], the ternary
complex accumulates. This is the basis for the theory of the GTP shift
for agonist binding (4). In any process involving more than a single
step, the intermediate species before the rate-determining step
accumulates; therefore, it can reasonably be supposed that when GTP
levels are very low, the rate at which adenylyl cyclase is activated
becomes directly dependent on the rate at which the ternary complex
breaks down. In the current report, however, we performed a complete
kinetic analysis over the full range of GTP concentrations from very
low to saturating concentrations of GTP. A complete theoretical
analysis for the relationship between GTP concentration and adenylyl
cyclase activity is given for the shuttle model of cyclase activation
(Appendix I). However, although the overall equation is somewhat
different when developed for a precoupled model (18), the conclusion
that the rate of breakdown of the ternary complex can be inferred from
the adenylyl cyclase activity is the same in both cases. It should be
pointed out that the exact form of the relationship between adenylyl
cyclase activity and GTP concentration is not crucial to the test of
the two-state model performed in this report. The solution for that
relationship was derived using a Gs-to-C shuttle model as
the primary mechanism for adenylyl cyclase activation because it is
interesting in its own right and because it allows the mathematical
demonstration that as [GTP] 0, the adenylyl cyclase activity is
directly proportional to the rate of ternary complex breakdown
(Appendix I).
Experimental conditions were chosen such that similar levels of
HRGtotal were used to stimulate adenylyl cyclase activity for four different 2-adrenergic agonists: the strong
agonists epinephrine and isoproterenol, the moderate agonist fenoterol, and the weak agonist dobutamine. If a strict two-state model were manifest for receptor conformation, then similar levels of HR*G are
predicted for each agonist when examining adenylyl cyclase activity
under limiting GTP concentrations. Appendix II provides the formal
thermodynamic justification of the statement above by showing that it
is possible to use GTP shift binding data to calculate the minimum
percentage of HRGtotal that must be in the HR*G state. Fig.
3 reveals the similarity in the HR*G levels between the agonists and
shows that the weak agonist dobutamine is capable of producing >80%
of the HR*G complexes as epinephrine under saturating agonist
conditions. Because the percentage is very high even for the weak
agonist dobutamine, the subsequent arguments are facilitated.
If the properties of the ternary complex depended proportionately only
on the fraction in the active HR*G form, then there should be little
difference between the ternary complex that included the weak agonist
dobutamine, the stronger agonist fenoterol, and the very strong
agonists epinephrine and isoproterenol. It would then be expected that
the rate of HRGtotal breakdown would be similar for all of
the agonists. Significant differences in the rates of breakdown would
demonstrate that a simple two-state model is inappropriate for this
process. Fig. 3 demonstrates that the relative rates of heterotrimer
breakdown for fenoterol and dobutamine are significantly different from
the minimum HR*G levels determined for each of the two agonists. This
suggests that the rate of HRGtotal breakdown is dependent
on the type of agonist, a finding that is inconsistent with a single
conformation for the activated state of the receptor.
It is unlikely that the agonist-dependent differences in the rate of
heterotrimer breakdown are due to 1) the ability of a particular
agonist to form the ternary complex or 2) the agonist-dependent variation in the rates of agonist association and dissociation to the
2-adrenergic receptor. Our experiments were designed to minimize the effects described above by bringing adenylyl cyclase activity to steady state during a 3-min incubation period before the
addition of labeled ATP and subsequent data collection. Therefore, problems resulting from the delay in reaching full activity on the
addition of stimulating agonist do not arise. Similarly, the preincubation ensures that agonist binding has also attained a steady
state before measurements are taken. Thus, the rates at which agonist
binds and dissociates from the receptor should also not be a
significant factor.
A possible explanation for these data that would preserve a strict
two-state model requires the selective activation of the inhibitory
heterotrimeric G protein, Gi. If Gi were
preferentially activated by dobutamine and fenoterol, then the reduced
adenylyl cyclase activity at low GTP, under stimulation by those
agonists, would be the result of Gi-mediated inhibition
rather than a slower rate of breakdown of the ternary complex. In the
current circumstances, this is unlikely because Gi-coupled
-adrenergic receptors have not been demonstrated in these cells. In
addition, the low level of -adrenergic receptors makes it unlikely
that a low-efficiency interaction between the -adrenergic receptor
and Gi could have a significant effect.
Our data therefore are not consistent with a strict two-state model as
described by Samama et al. (1). Our data do not, however,
invalidate the classic ternary complex model (4), but they do suggest
that receptor activation of Gs is more complicated than
agonist stabilization of an active R* state. Indeed, other investigators have suggested that each agonist stabilizes a unique set
of receptor conformations (known as the multistate model) with a range
of G protein-activating abilities (17, 18). However, caution should be
taken in extrapolating data from solubilized receptor systems and
mutant receptors to biological systems. It should also be noted that
although other models for receptor activation are possible, it may be
that a more realistic model of agonist/receptor/G protein interactions
might only complicate matters while only marginally improving the
resolution of the two-state model. In any case, these findings have
important implications not only in the study of the mechanisms involved
in hormone-activated receptors but also for future studies involving
the search pharmaceutical agents that may selectively activate
different forms of a receptor.
| |
Footnotes |
Received December 26, 1996; Accepted April 4, 1997
This work was supported by National Institutes of Health
Grant RR07710.
Send reprint requests to: Roger Barber, Ph.D., The
University of Texas-Houston Medical School, Department of Integrative
Biology, Pharmacology, and Physiology, 6431 Fannin, P.O. Box 20708, Houston, TX 77225-0334. E-mail:
rbarber{at}farmr1.med.uth.tmc.edu
| |
Appendix I |
This section describes the derivation of a complete kinetic
analysis of steady state adenylyl cyclase activity with respect to
varying GTP concentrations in terms of the Cassel-Selinger cycle (19).
The scheme shown below shows the cycle of Gs activation and
inactivation after receptor activation. In this version, a shuttle
model for G*s to adenylyl cyclase catalytic unit is
assumed, although an analysis of the process in terms of a precoupled
model for Gs and adenylyl cyclase would give almost
identical values for the estimates of the rate of breakdown from
experimental data. The description of active adenylyl cyclase is given
as the solution of a quartic equation; however, the quartic itself can
be factored into two quadratic equations, which simplifies matters
somewhat.
In the scheme above, HR is the agonist-occupied receptor,
Gs is the inactivated stimulatory G protein,
G*s is activated G protein, C is inactive adenylyl
cyclase, and G*sC is the active adenylyl cyclase
complex. The kinetic constants governing the cycle are represented by
k1, the rate of heterotrimer (HRG) formation;
k10, the rate of dissociation of an inactive heterotrimer; k2, the rate of active
heterotrimer (HRG) dissociation; k3, the rate of
adenylyl cyclase activation by activated G*s, and
k1, the rate of G*s and
G*sC inactivation via the intrinsic GTPase
mechanism [because adenylyl cyclase does not seem to act as a GAP for
G*s (20) k1, the rates for
G*s and G*sC are identical].
In the presence of GTP, we want to determine the steady state rate of
adenylyl cyclase, G*sC activity. From the
conservation of mass, the levels of R, Gs, and C are
defined as
![[G<SUB>s</SUB>]<SUB>total</SUB><IT>=</IT>[G<SUB>s</SUB>]<IT>+</IT>[HRG]<IT>+</IT>[G<SUP>*</SUP><SUB>s</SUB>]<IT>+</IT>[G<SUP>*</SUP><SUB>s</SUB>C]](/content/vol52/issue1/fulltext/144/img007.gif)
|
(10)
|
![[C]<SUB>total</SUB><IT>=</IT>[C]<IT>+</IT>[G<SUP>*</SUP><SUB>s</SUB>C]](/content/vol52/issue1/fulltext/144/img008.gif)
|
(11)
|
and under the experimental condition of saturating agonist
![[R]<SUB>total</SUB><IT>=</IT>[HR]<IT>+</IT>[HRG]](/content/vol52/issue1/fulltext/144/img009.gif)
|
(12)
|
At steady state, the amount of active
G*sC in the system is
![k<SUB>3</SUB>[G<SUP>*</SUP><SUB>s</SUB>][C]<IT>=k</IT><SUB>−<IT>1</IT></SUB>[G<SUP>*</SUP><SUB>s</SUB>C]](/content/vol52/issue1/fulltext/144/img010.gif)
|
(13)
|
and the amount of active G*s in the system is
![k<SUB>1</SUB>[HR][G<SUB>s</SUB>]<IT>=k</IT><SUB>−<IT>1</IT></SUB>([G<SUP>*</SUP><SUB>s</SUB>C]<IT>+</IT>[G<SUP>*</SUP><SUB>s</SUB>])<IT>+k</IT><SUB><IT>10</IT></SUB>[HRG]](/content/vol52/issue1/fulltext/144/img011.gif)
|
(14)
|
and the amount of [HRG] is
![k<SUB>1</SUB>[HR][G<SUB>s</SUB>]<IT>=</IT>(<IT>k</IT><SUB><IT>2</IT></SUB>[GTP]<IT>+k</IT><SUB><IT>10</IT></SUB>)[HRG]](/content/vol52/issue1/fulltext/144/img012.gif)
|
(15)
|
By substituting eq. 11 into eq. 13, we obtain
![[G<SUP>*</SUP><SUB>s</SUB>]<IT>=</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB>[G<SUP>*</SUP><SUB>s</SUB>C]</NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>([C]<SUB>total</SUB><IT>−</IT>[G<SUP>*</SUP><SUB>s</SUB>C])</DE></FR>](/content/vol52/issue1/fulltext/144/img013.gif)
|
(16)
|
Eqs. 12, 14, and 15 can be rewritten as
<IT>=k</IT><SUB>−<IT>1</IT></SUB>([G<SUP>*</SUP><SUB>s</SUB>C]<IT>+</IT>[G<SUP>*</SUP><SUB>s</SUB>])<IT>+k</IT><SUB><IT>10</IT></SUB>[HRG]](/content/vol52/issue1/fulltext/144/img014.gif)
|
(17)
|
and
![k<SUB>1</SUB>([R]<SUB>total</SUB><IT>−</IT>[HRG])[G<SUB>s</SUB>]<IT>=</IT>(<IT>k</IT><SUB><IT>2</IT></SUB>[GTP]<IT>+k</IT><SUB><IT>10</IT></SUB>)[HRG]](/content/vol52/issue1/fulltext/144/img015.gif)
|
(18)
|
Therefore,
![[HRG]<IT>=</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB>([G<SUP>*</SUP><SUB>s</SUB>C]<IT>+</IT>[G<SUP>*</SUP><SUB>s</SUB>])</NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR>](/content/vol52/issue1/fulltext/144/img016.gif)
|
(19)
|
and
![[G<SUB>s</SUB>]<IT>=</IT><FR><NU>(<IT>k</IT><SUB><IT>2</IT></SUB>[GTP]<IT>+k</IT><SUB><IT>10</IT></SUB>)[HRG]</NU><DE><IT>k</IT><SUB><IT>1</IT></SUB>([R]<SUB>total</SUB><IT>−</IT>[HRG])</DE></FR>](/content/vol52/issue1/fulltext/144/img018.gif)
|
(20)
|
Substitution of HRG from eq. 19 into eq. 20, after some
rearrangement, yields
![[G<SUB>s</SUB>]<IT>=</IT><FR><NU>(<IT>k</IT><SUB><IT>2</IT></SUB>[GTP]<IT>+k</IT><SUB><IT>10</IT></SUB>)<FENCE><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><FENCE>[G<SUP>*</SUP><SUB>s</SUB>C]<IT>+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB>[G<SUP>*</SUP><SUB>s</SUB>C]</NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>([C]<SUB>total</SUB><IT>−</IT>[G<SUP>*</SUP><SUB>s</SUB>C])</DE></FR></FENCE></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE></NU><DE><IT>k</IT><SUB><IT>1</IT></SUB><FENCE>[R]<SUB>total</SUB><IT>−</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><FENCE>[G<SUP>*</SUP><SUB>s</SUB>C]<IT>+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB>[G<SUP>*</SUP><SUB>s</SUB>C]</NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>([C]<SUB>total</SUB><IT>−</IT>[G<SUP>*</SUP><SUB>s</SUB>C])</DE></FR></FENCE></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE></DE></FR>](/content/vol52/issue1/fulltext/144/img019.gif)
|
(21)
|
Substitution of eq. 21 for [Gs], eq. 19, for [HRG]
and of eq. 16 for [G*s] into eq. 10 yields
![+<FR><NU>k<SUB>−<IT>1</IT></SUB><FENCE>[G<SUP>*</SUP><SUB>s</SUB>C]<IT>+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB>[G<SUP>*</SUP><SUB>s</SUB>C]</NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>([C]<SUB>total</SUB><IT>−</IT>[G<SUP>*</SUP><SUB>s</SUB>C])</DE></FR></FENCE></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR>](/content/vol52/issue1/fulltext/144/img021.gif)
|
(22)
|
By letting x = [G*sC] and
y = ([C]total [G*sC]), we obtain
![[G<SUB>s</SUB>]<SUB>total</SUB><IT>=</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><FENCE><IT>x+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><IT>x</IT></NU><DE><IT>k<SUB>3</SUB>y</IT></DE></FR></FENCE><FENCE><FR><NU><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]<IT>+k</IT><SUB><IT>10</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE></NU><DE><IT>k</IT><SUB><IT>1</IT></SUB><FENCE>[R]<SUB>total</SUB><IT>−</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><FENCE><IT>x+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><IT>x</IT></NU><DE><IT>k<SUB>3</SUB>y</IT></DE></FR></FENCE></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE></DE></FR>](/content/vol52/issue1/fulltext/144/img023.gif)
|
(23)
|
Therefore,
![=k<SUB>−<IT>1</IT></SUB><FENCE><IT>x+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><IT>x</IT></NU><DE><IT>k<SUB>3</SUB>y</IT></DE></FR></FENCE><FENCE><FR><NU><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]<IT>+k</IT><SUB><IT>10</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE><IT>+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR>](/content/vol52/issue1/fulltext/144/img026.gif)
|
(24)
|
After some rearrangement and writing
|
(25)
|
as z, we obtain
![<FR><NU>k<SUB>1</SUB>k<SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE>(<IT>xz</IT>)<SUP><IT>2</IT></SUP>](/content/vol52/issue1/fulltext/144/img030.gif)
|
(26)
|
Eq. 26 is a quadratic in xz and can be solved as a
quadratic to give xz, but
![xz=[G<SUP>*</SUP><SUB>s</SUB>C]<FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>([C]<SUB>total</SUB><IT>−</IT>[G<SUP>*</SUP><SUB>s</SUB>C])</DE></FR></FENCE><IT>=</IT>A,](/content/vol52/issue1/fulltext/144/img034.gif)
|
(27)
|
where A is the solution to the previous quadratic. Then
or
![[G<SUP>*</SUP><SUB>s</SUB>C]<SUP><IT>2</IT></SUP><IT>−</IT><FENCE>[C]<SUB>total</SUB><IT>+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB></DE></FR><IT>+</IT>A</FENCE>[G<SUP>*</SUP><SUB>s</SUB>C]<IT>+</IT>A[C]<SUB>total</SUB><IT>=0</IT>](/content/vol52/issue1/fulltext/144/img036.gif)
|
(28)
|
Relating the rate of heterotrimer breakdown to the
y-intercept of the Eadie-Hofstee plot.
From eq. 27, we
find that
![xz=[G<SUP>*</SUP><SUB>s</SUB>C]<FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>([C]<SUB>total</SUB><IT>−</IT>[G<SUP>*</SUP><SUB>s</SUB>C])</DE></FR></FENCE><IT>.</IT>](/content/vol52/issue1/fulltext/144/img037.gif)
|
(29)
|
When [GTP] 0, [G*sC] also approaches 0 and
![xz → [G<SUP>*</SUP><SUB>s</SUB>C]<FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>[C]<SUB>total</SUB></DE></FR></FENCE><IT>.</IT>](/content/vol52/issue1/fulltext/144/img038.gif)
|
(30)
|
Substituting this relationship for xz into eq. 26
yields
![<FR><NU>k<SUB>1</SUB>k<SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE><FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>[C]<SUB>total</SUB></DE></FR></FENCE><SUP><IT>2</IT></SUP>[G<SUP>*</SUP><SUB>s</SUB>C]<SUP><IT>2</IT></SUP>](/content/vol52/issue1/fulltext/144/img039.gif)
|
(31)
|
Because [GTP] 0, then
![1+<FR><NU>k<SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><IT> &Gtv; 1</IT>](/content/vol52/issue1/fulltext/144/img043.gif)
|
(32)
|
in eq. 31, which yields approximately:
![<FR><NU>k<SUB>1</SUB>k<SUP><IT>2</IT></SUP><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUP><IT>2</IT></SUP><SUB><IT>2</IT></SUB>[GTP]<SUP><IT>2</IT></SUP></DE></FR><FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>[C]<SUB>total</SUB></DE></FR></FENCE><SUP><IT>2</IT></SUP>[G<SUP>*</SUP><SUB>s</SUB>C]<SUP><IT>2</IT></SUP>](/content/vol52/issue1/fulltext/144/img044.gif)
|
(33)
|
Then, writing
![&thgr;=<FR><NU>[G<SUP>*</SUP><SUB>s</SUB>C]</NU><DE>[GTP]</DE></FR>](/content/vol52/issue1/fulltext/144/img048.gif)
|
(34)
|
and
in eq. 33, we obtain
![k<SUB>1</SUB>&phgr;<SUP>2</SUP>−{(k<SUB>1</SUB>[R]<SUB>total</SUB><IT>+k</IT><SUB><IT>10</IT></SUB>)<IT>+k</IT><SUB><IT>1</IT></SUB>[G<SUB>s</SUB>]<SUB>total</SUB>}<IT>&phgr;+k</IT><SUB><IT>1</IT></SUB>[R]<SUB>total</SUB>[G<SUB>s</SUB>]<SUB>total</SUB><IT>=0</IT>](/content/vol52/issue1/fulltext/144/img050.gif)
|
(35)
|
The solution to the quadratic eq. shown in eq. 35 is
![&phgr;=<FR><NU>k<SUB>1</SUB>([R]<SUB>total</SUB><IT>+</IT>[G<SUB>s</SUB>]<SUB>total</SUB>)<IT>+k<SUB>10</SUB>±</IT><RAD><RCD><AR><R><C>{<IT>k</IT><SUB><IT>1</IT></SUB>([R]<SUB>total</SUB><IT>+</IT>[G<SUB>s</SUB>]<SUB>total</SUB>)<IT>+k</IT><SUB><IT>10</IT></SUB>}<SUP><IT>2</IT></SUP></C></R><R><C><IT> −4k</IT><SUP><IT>2</IT></SUP><SUB><IT>1</IT></SUB>[R]<SUB>total</SUB>[G<SUB>s</SUB>]<SUB>total</SUB></C></R></AR></RCD></RAD></NU><DE><IT>2k</IT><SUB><IT>1</IT></SUB></DE></FR>](/content/vol52/issue1/fulltext/144/img051.gif)
|
(36)
|
Rearrangement yields
![&phgr;=<FR><NU>([R]<SUB>total</SUB><IT>+</IT>[G<SUB>s</SUB>]<SUB>total</SUB>)<IT>+</IT>(<IT>k<SUB>10</SUB>/k</IT><SUB><IT>1</IT></SUB>)<IT>±</IT><RAD><RCD><AR><R><C>{([R]<SUB>total</SUB><IT>+</IT>[G<SUB>s</SUB>]<SUB>total</SUB>)<IT>+</IT>(<IT>k<SUB>10</SUB>/k</IT><SUB><IT>1</IT></SUB>)<SUP><IT>2</IT></SUP></C></R><R><C><IT>−4</IT>[R]<SUB>total</SUB>[G<SUB>s</SUB>]<SUB>total</SUB></C></R></AR></RCD></RAD></NU><DE><IT>2</IT></DE></FR>](/content/vol52/issue1/fulltext/144/img052.gif)
|
(37)
|
The right side of eq. 37 is, in fact, the quadratic solution to
the formation of HRGtotal; see Experimental
Procedures where
|
(38)
|
Therefore,
and replacing 
yields
![<FR><NU>k<SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB></DE></FR><FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>[C]<SUB>total</SUB></DE></FR></FENCE><IT>&thgr;=</IT>[HRG]<SUB>total</SUB>](/content/vol52/issue1/fulltext/144/img055.gif)
|
(39)
|
We previously demonstrated for a Gs-to-C shuttle
mechanism for adenylyl cyclase activation that
k3 
k1 (16).
Therefore,
![<FENCE>1+<FR><NU>k<SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>[C]<SUB>total</SUB></DE></FR></FENCE><IT>→ 1</IT>](/content/vol52/issue1/fulltext/144/img056.gif)
|
(40)
|
and because [GTP] 0, eq. 39 simplifies to
![&thgr;=<FR><NU>[G<SUP>*</SUP><SUB>s</SUB>C]C<SUB>p</SUB></NU><DE>[GTP]</DE></FR><IT>=</IT><FR><NU><IT>k</IT><SUB><IT>2</IT></SUB>[HRG]<SUB>total</SUB>C<SUB>p</SUB></NU><DE><IT>k</IT><SUB>−<IT>1</IT></SUB></DE></FR>](/content/vol52/issue1/fulltext/144/img057.gif)
|
(41)
|
where Cp represents the fraction of adenylyl cyclase
activity dependent on the [Gs]total level
(i.e., when [Gs]total < [C]total, then Cp is proportional to
[Gs]total, whereas when
[Gs]total > [C]total, then
Cp is proportional to [C]total).
| |
Appendix II |
The thermodynamics of receptor/G protein/agonist interactions in
terms of the two-state model.
In this section, we use the
two-state model to calculate the minimum fraction of receptors that
must be in the R* state in the ternary complex to achieve the measured
GTP shifts. If a receptor exists in two distinct conformational states
that have different affinities for a ligand (Ka
for state A and Kb for state B), then the
measured dissociation constant (Kd) for the
ligand is given by Kd = Kafa + Kbfb, where
fa is the fraction of receptor in state A in the
receptor/ligand complex, and fb is the fraction of
receptor in state B in the ligand/receptor complex. Because this is a
two-state model, fa plus fb is equal to
unity. The above equations is derived from the first principles in
Appendix III.
If we adhere strictly to the two-state model, then when a ligand binds
to a given receptor with increased affinity as a result of a change in
conditions, according to the above equation, which is completely
general, all changes in Kd must be interpreted
in terms of changes in fa and fb, because
Ka and Kb refer to the binding to the two defined states.
In the absence of GTP, the -adrenergic receptor forms a ternary
complex with Gs and agonist (4). In this ternary complex, the affinity of the agonist for the receptor is increased relative to
its binding in the presence of GTP when ternary complex does not
accumulate. If B is the high affinity form of the receptor, then
Kb < Ka, and the greater
measured affinity for ligand in the absence of GTP requires there be a
greater contribution of the B form to the ternary complex. In fact, the
measured greater affinity places limits on the fraction of each
conformation that is thermodynamically allowable.
If Kd1 is the dissociation constant of ligand
from the receptor alone and Kd2 is the
dissociation constant from the ternary complex, then we may write
Kd1 = Kafa1 + Kbfb1 and Kd2 = Kafa2 + Kbfb2. Noting that fa1 = 1 fb1 and
fa2 = 1 fb2, these two
equations may be combined and rearranged to give
|
(42)
|
The minimum value for fb2 occurs when
Kb and fb1 approach zero.
Under these circumstances,
|
(43)
|
Any other (positive) values for Kb and
fb1 give a greater value for fb2.
It can be seen by inspection that any significant difference in
Kd1 and Kd2 must lead to
large proportion of receptors (>0.9) in the fb2
state. In the specific case of this report, even the rather weak
agonist dobutamine requires that 80% (based on the differences in
affinity obtained using a nonlinear two-component curve fit with the
high and low values of the curve constant in addition to the low
affinity portion of the curve) of the ternary complex be in the HR*G
form (i.e., the active form) in terms of the two-state model.
| |
Appendix III |
In the following system, only a single dissociation constant is
observed provided the conversion time between the different state of
the receptor R 
R* is rapid compared with the time allowed for
equilibration of ligand binding.
In this scheme, the dissociation for agonist binding
KD is given as
![K<SUB>D</SUB>=<FR><NU>[Free receptor][H]</NU><DE>[Bound receptor]</DE></FR><IT>=</IT><FR><NU>([R<SUB>A</SUB>]<IT>+</IT>[R<SUB>B</SUB>])[H]</NU><DE>([HR<SUB>A</SUB>]<IT>+</IT>[HR<SUB>B</SUB>])</DE></FR>](/content/vol52/issue1/fulltext/144/img060.gif)
|
(45)
|
Because
![K<SUB>A</SUB>=<FR><NU>[R<SUB>A</SUB>][H]</NU><DE>[HR<SUB>A</SUB>]</DE></FR><IT>, K<SUB>B</SUB>=</IT><FR><NU>[R<SUB>B</SUB>][H]</NU><DE>[HR<SUB>B</SUB>]</DE></FR><IT>, </IT>and<IT> </IT>M<IT>=</IT><FR><NU>[HR<SUB>A</SUB>]</NU><DE>[HR<SUB>B</SUB>]</DE></FR>](/content/vol52/issue1/fulltext/144/img061.gif)
|
(46)
|
we can write
![K<SUB>D</SUB>=<FR><NU><FENCE><FR><NU>M[HR<SUB>B</SUB>]<IT>K</IT><SUB><IT>A</IT></SUB></NU><DE>[H]</DE></FR><IT>+</IT><FR><NU>[HR<SUB>B</SUB>]<IT>K</IT><SUB><IT>B</IT></SUB></NU><DE>[H]</DE></FR></FENCE>[H]</NU><DE>M[HR<SUB>B</SUB>]<IT>+</IT>[HR<SUB>B</SUB>]</DE></FR>](/content/vol52/issue1/fulltext/144/img062.gif)
|
(47)
|
where fHRA and
fHRB are, respectively, the fractions of
receptor in the A and B conformations after ligand has
bound.
| |
References |
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Copyright © 1997 by The American Society for Pharmacology and Experimental Therapeutics
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Summary
Introduction
Summary
![<FR><NU>[<IT>R</IT>]<SUB>total</SUB><IT>+</IT>[G]<SUB>total</SUB><IT>+K</IT><SUB>RG</SUB><IT>−</IT><RAD><RCD><IT> </IT><AR><R><C>([R]<SUB>total</SUB><IT>+</IT>[G]<SUB>total</SUB><IT>+K</IT><SUB>RG</SUB>)<SUP><IT>2</IT></SUP></C></R><R><C><IT> −4</IT>[R]<SUB>total</SUB>[G]<SUB>total</SUB></C></R></AR></RCD></RAD></NU><DE><IT>2</IT></DE></FR>](/content/vol52/issue1/fulltext/144/img003.gif)
![<IT>=</IT><FR><NU><IT>k</IT><SUB><IT>−1</IT></SUB>([G<SUP>*</SUP><SUB>s</SUB>C]<IT>+</IT>[(<IT>k</IT><SUB>−<IT>1</IT></SUB>[G<SUP>*</SUP><SUB>s</SUB>C])<IT>/</IT>(<IT>k</IT><SUB><IT>3</IT></SUB>([C]<SUB>total</SUB><IT>−</IT>[G<SUP>*</SUP><SUB>s</SUB>C]))])</NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR>](/content/vol52/issue1/fulltext/144/img017.gif)
![[G<SUB>s</SUB>]<SUB>total</SUB><IT>=</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><FENCE>[G<SUP>*</SUP><SUB>s</SUB>C]<IT>+</IT><FENCE><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB>[G<SUP>*</SUP><SUB>s</SUB>C]</NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>([C]<SUB>total</SUB><IT>−</IT>[G<SUP>*</SUP><SUB>s</SUB>C])</DE></FR></FENCE></FENCE><FENCE><FR><NU><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]<IT>+k</IT><SUB><IT>10</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE></NU><DE><IT>k</IT><SUB><IT>1</IT></SUB><FENCE>[R]<SUB>total</SUB><IT>−</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><FENCE>[G<SUP>*</SUP><SUB>s</SUB>C]<IT>+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB>[G<SUP>*</SUP><SUB>s</SUB>C]</NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>([C]<SUB>total</SUB><IT>−</IT>[G<SUP>*</SUP><SUB>s</SUB>C])</DE></FR></FENCE></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE></DE></FR>](/content/vol52/issue1/fulltext/144/img020.gif)
![+<FR><NU>k<SUB>−<IT>1</IT></SUB>[G<SUP>*</SUP><SUB>s</SUB>C]</NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>([C]<SUB>total</SUB><IT>−</IT>[G<SUP>*</SUP><SUB>s</SUB>C])</DE></FR><IT>+</IT>[G<SUP>*</SUP><SUB>s</SUB>C]](/content/vol52/issue1/fulltext/144/img022.gif)
![<IT>+</IT><FR><NU><IT>k</IT><SUB><IT>−1</IT></SUB><FENCE><IT>x+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><IT>x</IT></NU><DE><IT>k<SUB>3</SUB>y</IT></DE></FR></FENCE></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><IT>+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><IT>x</IT></NU><DE><IT>k<SUB>3</SUB>y</IT></DE></FR><IT>+x</IT>](/content/vol52/issue1/fulltext/144/img024.gif)
![k<SUB>1</SUB><FENCE>[R]<SUB>total</SUB><IT>−</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><FENCE><IT>x+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><IT>x</IT></NU><DE><IT>k<SUB>3</SUB>y</IT></DE></FR></FENCE></FENCE>[G<SUB>s</SUB>]<SUB>total</SUB>](/content/vol52/issue1/fulltext/144/img025.gif)
![·<FENCE>x+<FR><NU>k<SUB>−<IT>1</IT></SUB><IT>x</IT></NU><DE><IT>k<SUB>3</SUB>y</IT></DE></FR></FENCE><IT>k</IT><SUB><IT>1</IT></SUB><FENCE>[R]<SUB>total</SUB><IT>−</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><FENCE><IT>x+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><IT>x</IT></NU><DE><IT>k<SUB>3</SUB>y</IT></DE></FR></FENCE></FENCE>](/content/vol52/issue1/fulltext/144/img027.gif)
![+x<FENCE>1+<FR><NU>k<SUB>−<IT>1</IT></SUB></NU><DE><IT>k<SUB>3</SUB>y</IT></DE></FR></FENCE><IT>k</IT><SUB><IT>1</IT></SUB><FENCE>[R]<SUB>total</SUB><IT>−</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><FENCE><IT>x+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><IT>x</IT></NU><DE><IT>k<SUB>3</SUB>y</IT></DE></FR></FENCE></FENCE>](/content/vol52/issue1/fulltext/144/img028.gif)
![<IT>−</IT><FENCE><IT>k<SUB>−1</SUB>+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><IT>k</IT><SUB><IT>10</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><IT>+k</IT><SUB><IT>1</IT></SUB>[R]<SUB>total</SUB><FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE></FENCE>](/content/vol52/issue1/fulltext/144/img031.gif)
![<IT>+</IT><FR><NU><IT>k<SUB>1</SUB>k</IT><SUB><IT>−1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR>[G<SUB>s</SUB>]<SUB>total</SUB>}(<IT>xz</IT>)](/content/vol52/issue1/fulltext/144/img032.gif)
![<IT>+k</IT><SUB><IT>1</IT></SUB>[<IT>R</IT>]<SUB>total</SUB>[G<SUB>s</SUB>]<SUB>total</SUB><IT>=0</IT>](/content/vol52/issue1/fulltext/144/img033.gif)
<IT>=k<SUB>3</SUB></IT>A[C]<SUB>total</SUB><IT>−k<SUB>3</SUB></IT>A[G<SUP>*</SUP><SUB>s</SUB>C]](/content/vol52/issue1/fulltext/144/img035.gif)
![<IT> −</IT><FENCE><IT>k<SUB>−1</SUB>+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB><IT>k</IT><SUB><IT>10</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><IT>+k</IT><SUB><IT>1</IT></SUB>[R]<SUB>total</SUB><FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE></FENCE>](/content/vol52/issue1/fulltext/144/img040.gif)
![<IT> +</IT><FR><NU><IT>k<SUB>1</SUB>k</IT><SUB><IT>−1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><IT> </IT>[G<SUB>s</SUB>]<SUB>total</SUB>}<FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>[C]<SUB>total</SUB></DE></FR></FENCE>[G<SUP>*</SUP><SUB>s</SUB>C]](/content/vol52/issue1/fulltext/144/img041.gif)
![<IT> +k</IT><SUB><IT>1</IT></SUB>[<IT>R</IT>]<SUB>total</SUB>[G<SUB>s</SUB>]<SUB>total</SUB><IT>=0</IT>](/content/vol52/issue1/fulltext/144/img042.gif)
![<IT>−</IT><FENCE><FR><NU><IT>k<SUB>−1</SUB>k</IT><SUB><IT>10</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><IT>+</IT><FR><NU><IT>k<SUB>1</SUB>k</IT><SUB>−<IT>1</IT></SUB>[R]<SUB>total</SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR><IT>+</IT><FR><NU><IT>k<SUB>1</SUB>k</IT><SUB>−<IT>1</IT></SUB>[G<SUB>s</SUB>]<SUB>total</SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB>[GTP]</DE></FR></FENCE>](/content/vol52/issue1/fulltext/144/img045.gif)
![<FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB><IT>−1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>[C]<SUB>total</SUB></DE></FR></FENCE>[G<SUP>*</SUP><SUB>s</SUB>C]](/content/vol52/issue1/fulltext/144/img046.gif)
![<IT>+k</IT><SUB><IT>1</IT></SUB>[<IT>R</IT>]<SUB>total</SUB>[G<SUB>s</SUB>]<SUB>total</SUB><IT>=0</IT>](/content/vol52/issue1/fulltext/144/img047.gif)
![&phgr;=<FR><NU>k<SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>2</IT></SUB></DE></FR><FENCE><IT>1+</IT><FR><NU><IT>k</IT><SUB>−<IT>1</IT></SUB></NU><DE><IT>k</IT><SUB><IT>3</IT></SUB>[C]<SUB>total</SUB></DE></FR></FENCE><IT>&thgr;</IT>](/content/vol52/issue1/fulltext/144/img049.gif)
![&phgr;=[HRG]<SUB>total</SUB>](/content/vol52/issue1/fulltext/144/img054.gif)
