Chapter 7: Cubic Reaction Schemes. ATSM and HOTSM
Some general considerations about cubic models are described in sub-chapter 7.1, followed by the derivation and application of some cubic models, especially the allosteric two-state model (ATSM) and the homotropic two-state model (HOTSM).
7.1 Introduction to Cubic Models
7.1.1. Two Simple Cubic Models
We have now advanced to two two-state models that are essential for our conception of mechanisms in modulatory regulation at the receptor level. In both models, two ligands can interact via a receptive unit and the receptor may exist unliganded in two isomeric forms, a reactive or an active conformation. The two models are the allosteric two-state model (ATSM) and the homotropic two-state model (HOTSM) (Fig. 7.1). Two different (heterotropic) ligands act in the ATSM, while two identical (homotropic) ligands are active in the HOTSM. One could argue that a better name for the allosteric two-state model would be the ‘heterotropic two-state model’ (HETSM), but currently I will adhere to the name settled in the literature – the allosteric two-state model (ATSM).
Citation: Bindslev N 2008. DOI: 10.3402/bindslev.2008.10
© 2008 N Bindslev. This book and all matter and items published therein are distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License (http://creativecommons.org/licenses/by-nc/3.0/), permiting all non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The ATSM was first analyzed by Hall (2000) varying either the agonist or the modulator concentration separately. Although the concentration of the heterotropic ligands in ATSM may be varied simultaneously in a fixed ratio (see later), this possibility was not explored by Hall (2000).
The HOTSM was developed by Bindslev (2004). In this model, the agonist and the modulator concentration by necessity vary simultaneously, since agonists and modulators will be identical molecules.
As we shall see, when a mixture of agonist and modulator molecules with a fixed concentration ratio are employed for dose-responses in the functional form of ATSM, this model behaves as a HOTSM.
Both models are explored in more detail in the two papers mentioned above, and should be consulted for an in-depth understanding of the model characteristics and capabilities as tools for synagic analysis. Here I shall present some highlights and additional possibilities for the two models.
There are three principal conditions in both the ATSM and the HOTSM: (1) binding of one ligand, (2) binding of two ligands simultaneously, and (3) a conformational isomerization for activation. Reaction schemes to describe the possible combinations of these three conditions result in cubic networks with seven independent system constants, as shown in Fig. 1A + B.
The ATSM is a cubic combination of the intervention model (Chapter 2) and the cyclic two-state model (cTSM) (Chapter 5). The HOTSM is a cubic combination of the auto-intervention model (Chapter 3) and again the cTSM described in Chapter 5.
The ATSM and HOTSM are both parts of and thus included in the four-pane two-state model (FP-TSM) in Fig. 7.2. The designation of system constants in model-pair ATSM and HOTSM in Fig. 7.1 is transferred from the FP-TSM in Fig. 7.2 in order to maintain a consistency of terms for the ATSM and HOTSM with those of the mother-model, FP-TSM. Terms for the receptor complexes and the parameters for ATSM and HOTSM are listed in Table 7.1 and Box 7.1. Apostrophes and subscripts at certain parameter symbols secure the difference between the two models (compare panels A and B in Fig. 7.1 and Fig. 7.2).
Figure 7.2. ATSM and HOTSM as parts of the four-pane two-state model (FP-TSM). (A) Without system constants. (B) With system constants.
Table 7.1. Ligands, receptor conformations and parameters in ATSM and HOTSM|
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Compare Table 7.1 with Box 7.1.
Box 7.1. Receptor conformations and parameters in ATSM and HOTSM
ATSM
Acronyms are as listed here for the eight receptor conformations together with symbols for the seven independent parameters of ATSM in Fig. 7.1A:
Receptor conformations in ATSM
| R | unbound reactive receptor |
| RS | agonist-bound reactive receptor |
| R* | unbound active receptor |
| R*S | agonist-bound active receptor |
| MR | modulator-bound reactive receptor |
| MR* | modulator-bound isomerized receptor with activity |
| MRS | agonist- and modulator-bound reactive receptor |
| MR*S | agonist- and modulator-bound receptor with activity |
| L | isomerization constant–efficacy constant |
| A ss | equilibrium association constant at the primary site for agonist S |
| A mm | equilibrium association constant at the secondary site for modulator M |
| a | intrinsic efficacy constant with an agonist already bound |
| b | intrinsic efficacy constant with a modulator already bound |
| c | co-operativity coefficient for binding of a second ligand when a first ligand is already bound |
| d | co-operativity coefficient for binding of a second ligand when a first ligand is already bound to an active receptor or an intrinsic efficacy constant when two ligands are bound |
HOTSM
Acronyms are as listed here for the eight receptor conformations together with symbols for the seven independent parameters of HOTSM in Fig. 7.1B:
Receptor conformations in HOTSM
| R | unbound reactive receptor |
| RS | substrate-bound reactive receptor |
| R* | unbound active receptor |
| R*S | substrate-bound active receptor |
| SR | modulator-bound reactive receptor without activity |
| SR* | modulator-bound isomerized receptor with or without activity |
| SRS | substrate- and modulator-bound reactive receptor |
| SR*S | substrate- and modulator-bound receptor with or without activity |
| L | isomerization constant–efficacy constant |
| A ss | equilibrium association constant at the primary site for substrate S |
| A ms | equilibrium association constant at the secondary site for substrate S |
| a | intrinsic efficacy constant with a substrate already bound at ‘O’ |
| b’ | intrinsic efficacy constant with a substrate already bound at ‘M’ |
| c’ | co-operativity coefficient for binding of a second ligand when a first ligand is already bound |
| d’ | co-operativity coefficient for binding of a second ligand when a first ligand is already bound to either site of an active receptor or an intrinsic efficacy constant when two ligands are bound |
‘O’ stands for the orthosteric or primary binding site, and ‘M’ designates the modulatory or secondary binding site.All parameters are forward constants for both ATSM and HOTSM.
The ATSM and the HOTSM are close siblings. In fact, it may be said that the HOTSM is a special model variant of the ATSM, since from a functional aspect, as already indicated, for ATSM with mixes of the two different ligands, when their ratio of concentration is kept constant during application, the ATSM becomes a copy of the HOTSM. This aspect of the ATSM/HOTSM pair was analyzed by Bindslev (2004), and discussed further in Section 7.8.1. However, formulation of occupancy in the two models results in completely different expressions (see Section 7.8.2).
7.1.2. A Promising Model for the Analysis of Modulator Action
Hall's ATSM is a clear choice for analyzing modulator effects and will probably be exploited in the coming years. ATSM is a modification of the ternary complex model of ‘allosteric’ interaction by Ehlert (1988). In the words of Hall (2000), the ATSM ‘provides a framework in which each of the receptor species in the ternary complex model can cause downstream functional effects with different efficacies’. In a fixed concentration ratio mode, both ATSM and HOTSM have three plateaus of effect, and the models predict that an allosteric ligand which affect the affinity but not the efficacy of a primary ligand will only influence a second plateau of the model's dose-response curves, and neither its third plateau at high ligand concentrations nor its initial first plateau before ligand application (see later in Fig. 7.27). It is only modulator ligands with an efficacy constants L different from unity that can affect the third level of function induced by a primary ligand.
Thus far, only sporadic implementation of the ATSM has been reported (see, e.g., May et al. 2004; Jensen & Spalding 2004; Franco et al. 2006; Langmead & Christopoulos 2006; Ehlert & Griffin 2008; and partially by Hoare et al. 2008). However, with the accelerated development of allosteric drugs and their clinical application (Table 7.2) (Gao & Jacobson 2006, sub-chapter 7.12), the relevance of ATSM as an analytical tool justifies a more extended examination of this model. Therefore, in Chapter 7, I will scrutinize the ATSM and its sibling the HOTSM for some general principles and some details.
Table 7.2. Examples of receptors for which there is development of allosteric drugs for better therapy (see also Raddatz et al. 2007)|
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*See also Baggio & Drucker (2007), especially pp. 2146–2149.
*Small peptide ago-allosteric modulator for a GPCR family B member.
7.1.3. Other Cubic Models and Other Models
Other reaction schemes have been presented which also result in cubic models when based on the three principal conditions mentioned above. For instance, for G protein coupled receptors (GPCRs), a cubic ternary-complex model (CTCM), was scrutinized by Weiss and co-workers (Weiss et al. 1996a b c). On its development see Fig. 7.3. In the CTCM it is merely the activated receptor complexes when coupled to G protein (GR*) that are considered functional in the transduction pathway (arrows in Fig. 7.4A). Nevertheless, even in the absence of agonists but with functionality linked to G proteins coupled to the receptor, the CTCM includes the possibilities for simulation of both spontaneous activity and inverse agonism, as inherent in ATSM and HOTSM.
To date, the CTCM has been implemented by few research groups (e.g., Bruheim et al. 2003; Monczor et al. 2003; Fitzsimmons et al. 2004; O'Brien et al. 2004; Pineyro et al. 2005).
Adding another ligand to the CTCM scheme, that is, operating with three different ligands, but baring simultaneity in binding, i.e., excluding tri-ligand binding, was also considered by Weiss and co-workers (1996a) and visualized in their Fig. 5. Other variations without explicit conformational isomerization have been suggested on the theme of cubic complex modeling, but with three ligands interacting by simultaneous binding (Marvizon & Baudry 1993; Kukkonen et al. 2001).
A model more complete than the CTCM for GPCRs was suggested by Christopoulos et al. (1998), and again by Christopoulos & Kenakin (2002), as before, requesting G protein coupling for activity and including spontaneous activity, inverse agonism, but now with tri-ligand binding, the same as simultaneous binding of three different ligands including (1) an agonist, (2) the G protein, and (3) a modulator molecule. It is a so-called cubic quaternary-complex model (CQCM) (Fig. 7.5A), with four principal conditions and still, as mentioned, implicating both inverse agonism and spontaneous activity in the presence of bound G protein. The CQCM may alternatively be represented as a cube-in-a-cube (Fig 7.5B). The cube-in-a-cube was discussed in a different setting as a ‘hyper-cube’ model with four ligands, that is, replacing the possibility of conformational isomerization by one extra ligand binding, leading to a pentanary-complex model (van Rijn & Willems-van Bree 2004).
The CTCM and its offspring, the CQCM, are further characterized in sub-chapter 7.11.
7.1.4. Contents of Chapter 7
As usual, both the ATSM and the HOTSM are analyzed for their characteristics related to binding studies (sub-chapters 7.2 and 7.5), as well as functional experimentation (sub-chapters 7.3 and 7.6). Further aspects on HOTSM are in sub-chapter 7.7. Overlap between ATSM and HOTSM is considered and visualized in sub-chapter 7.8.
A combination and extension of ATSM and HOTSM into the FP-TSM (Fig. 7.2), and this model's possibilities for simulating synagics are considered in sub-chapter 7.9. A subject related to the FP-TSM, the so-called OFCOR principle, is looked into in sub-chapter 7.10. Differences and similarities between ATSM/HOTSM and other cubic models are debated in sub-chapter 7.11. Finally, sub-chapter 7.12 exposes how a 10-year old market for allosteric drugs in therapy is longing for the implementation of a two-state model à la Hall's ATSM. For an overview, together with other models, some cubic models are listed in Table 7.3.
Table 7.3. Some examples of one-, two- (dimer), or multi-sited models in one-, two-, or three states involving one-, two-, or more ligands-and some in cubic representations (See also, for example Casadó et al. 2007)|
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7.2. Aspects for Binding with the ATSM
7.2.1. Basics of the Model
The ATSM by Hall (2000) is shown in Fig. 7.1A. Receptor conformations and parameters for the ATSM are given in Table 7.1 and Box 7.1. A distribution equation for the binding aspect of ATSM with ao as actual occupancy is:
Compare Eq. 7.1 with the corrected Hall Eq. 3 (Hall 2000, Erratum in: Mol Pharmacol 59: 161, 2001). In Hall's analysis it is the concentration of either an agonist or a modulator that is varied as the independent variable.
7.2.2. Variation in Single Parameters
Variations in single model parameters for the binding aspect of ATSM are shown in Fig. 7.6 for five different values of each parameter while increasing the agonist concentration.
7.2.3. Conclusion on Aspects of Binding as Simulated with the ATSM
The ATSM can model the effects of a modulator molecule on reduced as well as enhanced binding of an orthosteric ligand by changes in apparent affinity. The ATSM is well suited for a simulation of a modulator molecule that can either increase or decrease (a) binding of a radio-ligand, and it can also imitate (b) positive or negative effect on binding affinity of a neutral ant-agonist, and (c) positive or negative effects on binding affinity of an inverse agonist. All these effects can be observed in laboratory experiments with agonist, ant-agonist, and inverse agonist both as radio-ligands and non-labeled compounds. For binding, compare Fig. 7.7A with Hall's Fig. 3a, Fig. 7.7C with Hall's Fig. 4a, and Fig. 7.8A with Hall's Fig. 5a.
7.3. Aspects for Function with the ATSM
7.3.1 Distribution Equation
The conservation principle for function in the ATSM (Fig. 7.1A) may be formulated in a distribution equation given here as:
Symbols used are listed in Box 7.1 and Table 7.1. Compare Eq. 7.2 with Hall's equivalent expression (Hall 2000, Eq. 10). As for binding, in Hall's analysis it is the concentration of either an agonist or a modulator that is varied as independent variable, not concomitantly.
7.3.2 Variation in Single Parameters
Examples of variation in single model parameters for the functional ATSM (listed in Box 7.1) are shown in Fig. 7.9 for increasing concentrations of the agonist. See also the legend to Fig. 7.9 for further details.
7.3.3. Conclusion on Functional Aspects as Simulated with the ATSM
Observing the effects of parameter variation in the ATSM demonstrates how ATSM can simultaneously explain both positive and negative heterotropic modulation of function; by variation in b, c and d. For function, compare Fig. 7.10A + C with Hall's Fig. 6a,b, Fig. 7.11A + C with Hall's Fig. 7a,b, and Fig. 7.12A + C with Hall's Fig. 8a,c. In Fig. 7.13, there are plots of the functional ATSM for parameter L below and above 1/10 with modulator concentration [M] varying in five steps. Notice the spontaneous activity in plots of Fig. 7.13C + D for L at 0.2.
7.4 Further Aspects for the ATSM
Hall (2000) has analyzed the behavior of the ATSM for binding and function when either the agonist concentration or the modulator concentration varies as the independent variable. Some aspects on his analyses are presented and debated below.
7.4.1. Co-agonism
Co-agonism is seen for ligand-gated receptor channels (LGC) where a simultaneous presence of two different ligands is necessary to open the channel (Johnson & Ascher 1987; Dingledine et al. 1990; Marvizon & Baudry 1993; Corsi et al. 1996, Hall 2000). Other systems have been suggested with co-agonism, for instance, GABA-A channel receptors (Chang & Weiss 1999; Rusch et al. 2004; Rusch & Forman 2005).1
7.4.2. Discrepancy of Modulator Effects on Binding and Activity
In the ATSM with the right selection of parameters, as shown by Hall (2000), the occupancy with primary ligand S in a double-sited receptive unit can be almost independent of the concentration of an allosteric ligand, ranging over several orders of magnitude. Thus, the activity of the receptive unit may vary with changing concentrations of the allosteric ligand M, [M] (Fig. 7.13A and Hall's Fig 10a) with parameters a=10,000, b=1, c=8.5, d=0.03 and L=0.001, while the binding of S is almost constant for certain parameter values over an extended span of concentrations for [M] (Fig. 7.14A and Hall's Fig. 10b). This is an illustration of the situation for allosteric compound CPCCOEt at the human mGluR1b receptor, where there is no detectable change in glutamate binding for changing concentrations of CPCCOEt, while the activity changes with the change in [CPCCOEt] (Litschig et al. 1999; Hall 2000).
However, for this to occur with the ATSM, demands are high on selected parameters for the ATSM. Parameter L must be ≤1/1000, a must be ≥10,000, and concomitantly only narrow value ranges for d and c will qualify. The span for the right selection of parameter values is minute, as already indicated in Fig. 7.14B. Additionally, observe for instance the effects of minute variations of parameter d in Fig. 7.14C. The ATSM parameter values have to be within extremely narrow ranges to follow the action of CPCCOEt, therefore the above explanation for the behavior of CPCCOEt does not appear the most likely one, although it is a possibility.
7.5. Aspects for Binding with the HOTSM
7.5.1 A Distribution Equation for Occupancy in HOTSM
The HOTSM is shown in Fig. 7.1B, and its parameters and receptor conformations are listed in Table 7.1 and Box 7.1. A distribution equation for receptor occupancy in HOTSM based on the conservation principle must include factorization of receptor conformations where two identical ligands are bound at the same time.
We may list two slightly different occupancy models for the homotropic two-state reaction scheme. One in which we double the value of conformations in the reaction scheme with due respect to the fact that in conformations with two ligands bound, SRS and SR*S, they count twice. This first model merely includes the possible conformations. Therefore, the binding version of HOTSM is formulated with ao as actual occupancy:
The second formulation of binding in HOTSM further takes into account that all conformations have a double possibility of binding ligands, therefore binding is related to a double set of conformations by simply doubling all conformations in the denominator of the distribution equation equal to all possible binding situations (see, e.g., Cornish-Bowden 2004, Eq. 11.30 or Cornish-Bowden 1995, Eq. 9.27).2 This distribution equation is thus given by:
Note that in Eqs. 7.3 and 7.4, some system constants now appear with an apostrophe in order to differentiate them from similar parameters in the ATSM. Further, here for HOTSM, the association constant A ms replaces A mm of the ATSM.
A general form of Eq. 7.4 is given by Cornish-Bowden (2004), Eq. 11.32, 1995, Eq. 9.29) and a variant form is derived by Kurganow (1982), Eq. 3.41). Compare this with the equations for the random reaction scheme for binding in Chapter 6–Eq. 6.4 for two ligands and Eq. 6.8 for n ligands.
7.5.2. Effects of Variation in Single Parameters of Binding-HOTSM
A preliminary analysis of varying single parameters in the occupancy version of the HOTSM has been carried out (Bindslev 2004). Meanwhile in binding, this analysis did not take into account that some of the receptor conformations have to count twice when two agonist molecules are bound simultaneously, and as a further possibility that all bound conformations should be related relative to all possible conformation multiplied by a factor 2, as in Eq. 7.4 (see the arguments for Eq. 7.4 above).
An analysis of these aspects of the HOTSM in binding is presented here. A demonstration of the effects of varying single parameters in occupancy-HOTSM by the two principles in Eqs. 7.3 and 7.4 are shown for parameters b’, c’, d’, L, A ss, and A ms in Figs. 7.15 16 17 18 19 20 and detailed comments may be found in the figure legends.
With the extension in Eq. 7.4 compared to Eq. 7.3, the theory predicts a plateau level at 50% occupancy for certain parameter values, shown for b’ in Fig. 7.15, for c’ in Fig. 7.16, for A ss in Fig. 7.19, and for A ms in Fig. 7.20. However, in spite of the modification of Eq. 7.3 to Eq. 7.4, the introduced ‘flexibility’ of a plateau at 50% is not worth much in binding displacement experiments, which do not seem restrained to a plateau phase at exactly 50% occupancy; though, it may reasonably be argued to include this correction for the distribution equation of the binding-HOTSM.
7.5.3 Conclusion on Aspects of Binding as Simulated with the HOTSM
Positive co-operativity in occupancy can be simulated with HOTSM. A variant form of the HOTSM can describe terraced concentration binding curves, but merely terraced for binding around 50% occupancy. Also, compare these 50% plateaus with similar published plateaus for the square geometry in the KNF model by Koshland et al. (1966).
How is binding in HOTSM different from binding in ATSM? Binding in the HOTSM includes double occupancy with co-lateral binding, whereas in the ATSM co-lateral binding is excluded (compare plots in Figs. 7.15 7.16 7.17 7.18 7.19 7.20 with plots in Fig. 7.6).
The positive co-operativity of the HOTSM is insufficient for positive co-operativity in, for instance, hemoglobin that operates with four binding sites. Here, the HOTSM needs to be developed further.
7.6. Aspects for Function with the HOTSM
7.6.1 Simulation with HOTSM
The functional form of the homotropic two-state reaction model has several and surprising possible simulations available (Bindslev 2004). To illustrate, a research example, that is discussed in more detail in sub-chapter 7.7, is brought up here in the introduction (section 7.6.2), for which the HOTSM seems relevant in simulation of experimental self-inhibited and self-enhanced effects as reverse bell-shaped dose-responses.
Intrinsic self-enhancement of oxygen binding to hemoglobin following a deviation from the ordinarily hyperbolic load relation is not relevant for the functional form of HOTSM, while self-inhibition in the form of substrate inhibition, as described in Chapter 3, is relevant. The former is designated as positive co-operative, while the latter may be denoted as negative co-operative. However, since substrate inhibition can result in bell-shaped dose-responses (see Figs. 3.2 and 3.3), a designation such as ‘auto-ant-agonism’, ‘auto-inhibition’, or ‘negative auto-intervention’ for the decaying leg of bell-shaped dose-responses are more valid than ‘negative co-operativity’ because negative co-operativity is usually a reference to merely shallow dose-responses (Chapter 15) (Bindslev 2004). A general term when assuming genuine allostery which covers all the observed deviations from simple load relations, even reversed bell-shaped dose-responses, is auto-modulation. We may operate with positive and negative auto-modulation. HOTSM is well suited to simulate both positive and negative functional auto-modulation, including, as we shall see, reverse bell-shaped and reverse terraced dose-response relationships.
7.6.2 Examples of Reverse Bell-shaped Dose-responses
Due to significant spontaneous activity, typically in model systems of transfected cell, it is possible to observe reverse bell-shaped dose responses (Migeon & Nathanson 1994; Michal et al. 2001; Christopoulos et al. 2001; Accomazzo et al. 2002; Nasman et al. 2002; Hornigold et al. 2003; Holmqvist et al. 2005). These reverse bell-shaped dose-responses are not necessarily explained, as usual, by a G protein signal-bifurcation between for instance Gs and Gi complexes, since the bell-shaped dose-response may be present even after eliminating the function of Gi proteins by treatment with PTX of the receptor-transfected cells (Hornigold et al. 2003). Meanwhile, the observed reverse bell-shaped behavior is easily modeled by the HOTSM, suggesting an intrinsic receptor modulation as explanation. Thus, the HOTSM may also be used as an analytical tool for reverse bell-shaped dose-responses. This aspect of the HOTSM together with its inherent reversed terraced dose-responses is further discussed in sub-chapter 7.7. Models are discussed in Section 7.11.3 for the reverse bell-shaped dose-response of GPCRs where only a single type of G protein is present.
7.6.3. The Basics of HOTSM
The HOTSM is shown in Fig. 7.1B. It is a combination of the auto-intervention model described in Chapter 3 and the cTSM presented in Chapter 5. The receptor conformations and parameters for HOTSM are listed in Table 7.1 and Box 7.1, and its distribution equation for function can be formulated as:
see Bindslev (2004), Eq. 2). Again, note the parameters with apostrophe in order to differentiate them from similar parameters in the ATSM. The association constant A ms replaces the A mm of ATSM.
7.6.4. Variation in Single Parameters
In contrast to the ATSM, both concentrations of ligands at either binding site are varied simultaneously as they change concomitantly. This is compulsory for the HOTSM. I shall return to how this can be mimicked in the ATSM in sub-chapter 7.8.
For the functional form of the HOTSM, variations in parameters b’, c’, d’, L, A ss, and A ms are shown in Figs. 7.21 22 23 24 25 26 . Details of the parameter variations are commented on in the figure legends.
7.6.5 Conclusion on Functional Aspects as Simulated with the HOTSM
The functional HOTSM is a versatile tool for simulation of single-ligand dose-responses rendering load-deviant curves. Experimental curve forms that may be simulated with HOTSM are summarized in Fig. 7.27. In the past, such load-deviant curve forms of experimental synagics triggered use of the Hill and sums of the Hill equation as analytical tools (e.g., Bronnikov et al. 1999; Accomazzo et al. 2002; Hornigold et al. 2003). The HOTSM offers a mechanistic approach to the analysis, while the Hillian approach is nearly always a pure mathematical description lacking any mechanistic relevance (Chapter 10). Hence, future modeling ought to consider variations on the themes of the HOTSM, replacing versions of the Hill equation.Fig. 7.28
7.7. Further Aspects for the HOTSM
With the right combination of parameter values, the functional HOTSM possesses both bell-shaped and terraced as well as reverse bell-shaped and reverse terraced relationships (Fig. 7.27). A prerequisite for observing inverse agonism, reverse terraced curves, and reverse bell-shaped dose-response relations is spontaneous activity which is detectable for L>1/100. Parameter L is the sole system constant to determine the initial level of activity in both HOTSM (Fig. 7.23 and Fig. 7.24) and ATSM (Fig. 7.13C + D).
7.7.1. Reverse Bell-shaped Synagics
Hornigold et al. (2003) described a system only expressing a single receptor subtype, m3, coupled to producing cAMP as spontaneous activity and affected in a reverse bell-shaped dose-response relation by methacholine (Fig. 7.29A). Even after treatment with PTX in order to eliminate possible coupling to inhibitory G proteins, this system still displays reverse bell-shaped synagics (Fig. 7.29B). This excludes bifurcation via Gs and Gi as an explanation for the observed reverse bell-shaped dose-response relationship.
The concave bell-shaped dose-response in Fig. 7.29A can be simulated by the HOTSM (see curve in the figure) and may thus be interpreted as a receptor intrinsic modulation in line with the evidence of more than one binding site for ligands in functional muscarinic receptors (Huang & Ellis 2007).
Therefore, for a possible alternative explanation of the observed reverse bell-shaped dose-response curve, as mentioned above, the HOTSM is a simple, elegant, and sufficient analytical tool, as also pointed out by Bindslev (2004), Figs. 2 and 9).
7.7.2. The HOTSM Against the Hill Equation
The change between three plateau levels of the HOTSM in function (Fig. 7.27), and also the co-operative rise of the concentration-occupancy relation for the model in binding ( Figs. 7.15 7.16 7.17 7.18 7.19 7.20 ), are not as steep as dose-responses described by a single or a combination of two or more Hill equations. This is not surprising, as the whole analysis of the HOTSM is based on a system with only two binding sites. For a fair comparison, the HOTSM should be allowed to have as many sites as the Hill analyses in such situations (Bindslev 2004).
7.7.3. Pseudo-homotropic Two-state Model (PHOTSM)
*
A model similar to the extended two-state model for heterotropic interaction (ETCM) in Section 7.10.1, but now for homotropic behavior, co-operativity, has been put forward recently as two-state two-site model or a ‘two-state-dimer-model’ (Franco et al. 2005; Franco et al. 2007a b; Ehlert 2008). The former model, which is a fraction of the HOTSM, is capable of simulating reverse bell-shaped dose-response relations without an over-shoot (compare Fig. 3 in Franco et al. (2005) and Fig. 7.27). Obviously, the ‘two-state-dimer-model’ and the two-state two-site model are covered by the HOTSM. See Table 7.3 for more models.
7.8 Comparison between the ATSM and the HOTSM
From a functional viewpoint, the ATSM overlap with the HOTSM. From the aspect of occupancy, the two models cover separate issues and are very different.
7.8.1. Functional Studies
The HOTSM can be said to be a special version of the ATSM. The ATSM is congruent with the HOTSM when doses are applied from a mixture made of an orthosteric ligand and a modulatory ligand ( Figs. 7.21 7.22 7.23 7.24 7.25 7.26 ). As the concentration of ligands in the mixture is increased, a fixed proportion is kept between doses of the two ligands in the ATSM, just as for the HOTSM regime. Such an experiment with a primary ligand and a modulator in a fixed ratio will, in theory, follow the HOTSM when corrections are made for possible differences between affinity constants A mm and A ms, which in the model may simply be taken into account by multiplying either by a factor. Now the fixed ratio experiment has become a single-ligand study, and its analysis by the ATSM is identical to an analysis by the HOTSM. Aspects of the behavior of ATSM/HOTSM as system tools were presented by Bindslev (2004).
7.8.2. Binding Studies
The ATSM is a prime example for the simulation and analysis of dose-responses of occupancy in radio-ligand binding studies as affected by cold allosteric ligands, demonstrated by Hall (2000). In the HOTSM, application of a ligand that can bind to two sites simultaneously is different from the drug application described by Hall (compare Figs. 7.6 7.7 7.8 with Figs. 7.15 7.16 7.17 7.18 7.19 7.20 ).
To recapitulate, there is no feasible comparison between the ATSM and the HOTSM reaction schemes used in simulations for binding. The reaction scheme of ATSM can only bind ligands of interest for occupancy studies in a single site. Co-lateral binding is excluded. In the HOTSM, two ligands of interest can bind simultaneously in a co-lateral fashion by just adding one type of ligand. This clearly separates the two models from each other in binding studies.
7.9. Evaluation of FP-TSM as an Analytical Tool
7.9.1. Use of the FP-TSM Tool is Easy?
The FP-TSMs in Fig. 7.2 and 5.9 has 18 conformations and 17 independent system constants, and besides its genuine two-state isomerization, it involves two different ligands, an agonist or a substrate and a modulator, with mutual co-lateral binding. The possibility of co-lateral binding in FP-TSM differentiates it from the CQCM in Fig. 7.5.
Armed with some reasonable assumptions, the 17 constants can be reduced to around 12.
Still, to analyze the FP-TSM with 12 independent system constants sounds horrendous, but I believe it is only a matter of time before this model with unliganded spontaneous activity will be employed in the analysis of ligand–receptor interactions. How can I trust such a development? Well, consider all the efforts being put into drug design and drug development based and performed on the modeled dynamics of steric coordinates in ligand binding sites, and hold this against the immense degrees of freedom in possible alignment for receptive systems and their cognate ligands at an Angstrom scale. The possibilities are legio and will keep scientists and support floating for years – not necessarily with any foretold causal solutions.
7.9.2. Structure-activity-relationship (SAR)
The whole G protein-coupled receptor community eagerly awaits more details of the structural resolution of GPCRs, both for ligand binding sites and for the coupling between GPCRs and G proteins (Oldham et al. 2006; Kobilka & Scherkler 2008), as well as the interaction between GPCRs – G proteins and other proteins (Ferguson 2007), and looks enviously to the advancements of nearly 40 years in structural resolution of enzyme function. As the structural secrets are being revealed for transporters as well (Toyoshima 2007), porter-people are alert and expectations are high.
Meanwhile, even with recent promising breakthroughs in the structural resolution for receptors (Palechewski et al. 2000; Okada et al. 2002 2004; Palechewski 2006; Cherezov et al. 2007) and for several different transporters (Gouaux & MacKinnon 2005) as channels (Dutzler et al. 2002 2003; MacKinnon et al. 2003 2004; Unwin 2005; Jentsch et al. 2005a b; Miller 2006; Ramjeesingh et al. 2006; Jasti et al. 2007), pumps (Toyoshima et al. 2000; Obara et al. 2005; Jensen et al. 2006), exchangers (Dutzler et al. 2002 2003; Hunte et al. 2005; Jentsch et al. 2005a b; Accardi et al. 2006; Miller 2006; Nicoll et al. 2006; Nguitragool et al. 2006), ABC-transporters (Yu et al. 2003; Dawson & Locher 2006), and co-transporters (Yernool et al. 2004; Yamashita 2005; Indarte et al. 2007) leading the way to synthesis of new lead-compounds, dynamic possibilities in the binding pockets are still immense and overwhelming when it comes to the dynamic stereo-chemistry at a subatomic scale needed for the ultimate useful structure-activity-relationship (SAR).
7.9.3. The FP-TSM in SAR Analysis
In comparison with the efforts described in the preceding sections, to me it seems a modest task to develop the FP-TSM, which among other issues mechanistically may describe the reversal of auto-ant-agonism (reversal of negative auto-modulation or negative auto-intervention) by a classical competitive ant-agonist. An example of such reversal by a competitive ant-agonist has been observed for atropine at muscarinic receptors (Figure 30A) (Winding & Bindslev 1993), where atropine can reverse auto-inhibition induced by acetylcholine within one to two minutes, thus reactivating an auto-inhibited acetylcholine response of a cell membrane displayed muscarinic receptor pool (Fig. 7.30B).
In developing a scheme for this kind of ‘ant-agonism’, the terminology of designators in FP-TSM such as ‘competitive’ inhibitor, ‘non-competitive’ inhibitor, ‘interventor’, and ‘modulator’ may start to mingle. Atropine is a competitive ant-agonist in the classical sense (Brimblecombe 1974, Chapter 1; Carrijo et al. 1977; Hulme et al. 1978), but atropine will also appear as a non-competitive inhibitor or an interventor in intervention models, or a modulatory ligand in the ATSM and the FP-TSM. Although atropine is a ‘neutral’ ant-agonist, as it does not in itself evoke or prevent any activity in the absence of agonists or inverse agonists, it may have the characteristics of all four types of ligands mentioned above. However, atropine does not qualify as an ago-modulator, since it does not act on its own.
In the realm of G protein function, a start has been made with the implementation of a pre-FP-TSM (Downing et al. 2006). Furthermore, an alternative FP-TSM reaction scheme has been invoked for GABA-A channels (Rusch et al. 2004). The future looks bright for the FP-TSM as an analytical tool.
7.10. The Optimal-fixed-concentration-ratio (OFCOR)
7.10.1. Inhibition of Negative Auto-modification
In systems with negative auto-intervention or negative auto-modulation in the form of bell-shaped dose-responses at high agonist concentrations, this inhibition by the agonist itself may be counteracted by introducing a modulator or even a competitive ant-agonist, both with access to more than one agonist binding site.
As described above, a reactivation of such a negative auto-modulatory response was found for the ligand-pair acetylcholine/atropine in tracheal secretion (Fig. 7.30) (Winding & Bindslev 1993). There are several possible molecular explanations for this reactivation mechanism by atropine, one being competition at a secondary binding site with access for both inhibitory primary ligand (auto-inhibitory agonist acetylcholine) and a modulator (atropine). The FP-TSM is suited for such an explanation (see below).
Internalization of receptors as an explanation for the auto-inhibition in this system seems to be ruled out by the speedy reactivation within a minute or two by atropine (Fig. 7.30). Desensitization, converted to an attenuating dose-response, is another obvious possibility for the auto-inhibition by acetylcholine. Meanwhile, in this scenario, we have to explain how atropine behaves as a ‘competitive ant-agonist’ and how it resensitizes the auto-inhibition by acetylcholine in a time dependent manner.
As mentioned, the reactivation behavior can be addressed by a FP-OSM, which includes co-lateral binding at either site (Winding & Bindslev 1993). Ligand-relief of negative auto-modulation may of course also be simulated with the FP-TSM, which will be useful in case systems moreover comprise spontaneous activity.
7.10.2. Finding OFCORs
In case agonist concentrations in clinical treatment elicit auto-inhibition, thereby tempting a higher dose application with possible adverse effects, such auto-inhibition may be prevented by interventors, modulators, or ‘competitive ant-agonists’ with a ‘positive’ effect elicited from a secondary site. It will be possible to avoid adverse effects due to high concentrations of primary ligands that also elicit auto-inhibition by use of a single mixture of 'agonist' and 'auto-modulator' with an optimal concentration ratio of the two drugs. Of course, using such mixtures in dose-response experiments with optimum is the same as using OFCORs. The challenge is, of course, to find the right OFCORs for a given individual; his of hers drug-mixture.
Determining an optimal concentration ratio for experiments and treatments may be obtained by assaying the dose-response relations for and between the agonist and the modulator in several combinations of the two drugs. As an example, the OFCOR for a mixture was determined and verified in the study by Winding and Bindslev (1993) using a version of the FP-OSM. To date, a similar analysis using the FP-TSM is not in the literature.
7.11. More on Cubic Models and GPCR-related Models
7.11.1. The Extended Ternary Complex Model (ETCM)
In order to describe the dose-response relations obtained for GPCRs with constitutive activity and varied G protein levels, Samama and co-workers developed a model combining the original ternary-complex model equal to an intervention model (Chapter 2) for G proteins, with the cTSM (Chapter 5), in the simplest possible scheme (Samama et al. 1993; Lefkowitz et al. 1993). The ETCM for GPCRs is shown in Fig. 7.4B. Development of the ETCM was a stage on the road to the thermodynamically full model with modulators, including G proteins, and two-states combined in the CTCM (Fig. 7.3, Table 7.3) (Weiss et al. 1996a).
7.11.2. ATSM and CTCM and CQCM
When occupancy is considered, the ATSM (Hall 2000) for allosteric modulators is in principle identical to the CTCM for ‘modulation’ of GPCRs by G proteins (Fig. 7.1A and 7.4A) (Weiss et al. 1996a b). Examples of occupancy that cover both models are shown in Fig. 7.6.
The parallel occupancy between ATSM and CTCM is true also in simulation of experiments where the number of total receptors differs between radio-ligand agonist and radio-ligand ant-agonist data due to G protein-induced affinity changes (Kenakin 1997; Baker & Hill 2007; Ferguson 2007). Here, adjustments of parameter A mm come in to play.
On the other hand, in functional studies the two tools, ATSM and CTCM, behave somewhat differently (Weiss et al. 1996a c; Hall 2000). An illustration of this can be acknowledged by comparing the surface plot in 3-D panels of Figs. 23 and 31, where both concentrations of an agonist and the G protein as modulators are varied, as independent variables yielding the 3-D surface plots. Following the arrows in the 3-D concentration planes in Fig. 7.31A + B at an arbitrary concentration (expression level) of G protein exemplified with ‘1’ in Fig. 7.31A or at ‘100’ in Fig. 7.31B results in 2-D plots in A and B of Fig. 7.31, and should not be compared with the 2-D plots in Fig. 7.23 which are for the HOTSM. However, with simple modifications, the ATSM may as well be a model for GPCRs.
Experiments with varying expressions (concentrations) of G proteins are less common but have been performed, modeled, and discussed (Chidiac et al. 1996; Kenakin 1997; Tucek et al. 2001 2002; Kukkonen et al. 2001; Kukkonen 2004b).
With the CTCM it is possible to study GPCR dose-response relations with varying concentrations of the agonist as the independent variable and G proteins at different levels. The CTCM for G protein coupled to activated receptors, even in the absence of agonists GR*, obviously has possibilities of simulating spontaneous activity and inverse agonism, but in relation to non-G protein regulators, the CTCM is clearly unsatisfactory. Therefore, models such as the CTCM with an added third ligand are in demand and have been suggested as the CQCM (Christopoulos et al. 1998; Christopoulos & Kenakin 2002; May et al. 2004), but, to date, are not fully developed.
An amputated version of the CQCM with two G proteins, a GPCR, and an agonist yielding quaternary complexes3 but without an isomerization between two states, was implemented by Kukkonen et al. (2001) and Kukkonen (2004a), introducing a factor F controlling the level of receptors chelated by G proteins. Other simpler models, also without isomerization, including variations in G protein expression and a factor controlling maximal inhibition (MI) and maximal stimulation (MS) have been suggested (Tucek et al. 2002).
Needless to say, the CQCM illustrated in Fig. 7.5 should be developed sooner rather than later.
7.11.3. Models of ‘Ligand’ Numbers Matching ‘Receptor’ Numbers
In the latest development of models addressing the reverse bell-shaped dose-response relations for GPCRs, ‘dual effects’ (Jones et al. 1991; Kashihara et al. 1992; Migeon & Nathanson 1994; Vogel et al. 1995; Jakubik et al. 1996), two parallel approaches have been advocated for a limited G protein activation (Tucek et al. 2001 2002; Nasman et al. 2001; Kukkonen et al. 2001; Kukkonen 2004a). Tucek et al. (2002) and especially Kukkonen (2004a) give elegant descriptions of these models’ development and simulation capabilities; notwithstanding, arithmetical treatment in these two sources mutually overlook each others similarities in analysis.
Following Kukkonen, three principal models are invoked. The shuttling model, earlier described as a mobile receptor model (Jacobs & Cuatrecasas 1976) or collision coupling model (Tolkovsky & Levitzki 1978 1981), the complexing model (Chidiac 1998), and the precoupled model (Neubig 1994; Kukkonen 2004a). The latter comes in two versions; a precoupled-dependent and a precoupled-independent model. Somewhat different models on precoupling were also presented by Jacobs and Cuatrecassas (1976) and by Tolkovsky and Levitzki (1978).
In the recent models, the variation in a limited number of single type G protein that can be activated, is accomplished by a coefficient resembling the intrinsic activity factor of Ariens (1954) both in the Czech and the Swedish approach (Tucek et al. 2002; Kukkonen 2004a b). Thus, the limited number of active G proteins and the ratio between Gs and Gi proteins result in simulations that can match the observed reverse bell-shaped dose-responses. The simulated ‘spontaneous’ activity at 100% was obtained experimentally by forskolin stimulation.
7.11.4. Models with More than One-state for GPCRs
Model approaches to cover multi-states in GPCRs were recently presented, Table 7.3. For instance, one by Leff and co-workers (Leff et al. 1997; Scaramellini & Leff 1998 2002), another by Pin and co-workers (Parmentier et al. 2002) covering both two-states of a binding domain (see also Rovira et al. 2008) and two-state of an effector domain, and a third by Giraldo (2004). The models by Leff et al. and Giraldo both play on the theme of combining two cTSM for the sake of simulating GPCR functional behavior. These models operate with three or more states for the unliganded receptive unit, and may be elaborated further into multi-state models – a project left for the reader with some help from a recently published four-state model (Ehlert &Griffin 2008). A double-cubic model encompassing two states for both the receptor and its effector was already suggested in the 1970s (de Haen 1976).
7.11.5. Models for Binding with Dimerization as Isomerization
Dimerization of receptive units may lead to an allosteric interaction between binding sites in the dimer complex, thus resembling the switch between a reactive and an active state for a double-sited receptor. Recently, Durroux (2005) has suggested different and attractive models for such dimerization with ‘allosteric’ interaction, including cubic reaction schemes and four-pane two-state-like models. It will be interesting to follow development of Durroux’ models as they appear most relevant for a description and possible simulation of activity in dimerized GPCRs.
7.12. Use of Modulators in Therapy
7.12.1. Benefits of Modulators Compared with Orthosteric Ligands
An early example of modulators used in therapy with great success and few side-effects is the treatment of anxiety, depression, epilepsy, and sleeplessness with benzodiazepines, which started more than forty years ago. Nevertheless, the clinical aspect of the use of modulators or allosteric compounds is a relatively new issue, and has been advocated since the mid 1990s (Proska & Tucek 1994; Tucek & Proska 1995, Birdsall et al. 1995; Lazareno & Birdsall 1995). Some of the advantages of using modulator compounds instead of the natural endogenous agonist or artificial agonists with receptor subtype specificity can be listed as follows (Birdsall & Lazareno 2005):
(a) As the function of a modulator to a great extent is dependent on the presence and action of the physiological signal molecule, it is likely to mimic normal biological effects. The modulator is playing on the activity of an endogenous primary ligand.
(b) The enhancement of natural agonist activation or inhibition by modulators is less likely to reach adverse effects as for agonist drugs, even at high concentrations of the modulator compound, and therefore prolonged stimulation with high modulator concentrations may be induced without adverse effects as often seen with high concentrations of developed agonist drugs. There is ‘ceiling’ and ‘tune down’.
Besides reuptake and decay inhibitors, some of the expected therapeutic advantages with use of modulatory drugs may also be obtained with partial and partial inverse agonists (Fisas et al. 2006).
In clinical work with modulators, it is the functional aspect that is in focus. The picture of modulatory drug therapy has developed rapidly within the last few years for several receptor sub-families, and many developed drugs are in various phases of evaluation for introduction to the clinics (Table 7.2) (Gao & Jacobson 2006). Examples of biotech firms that develop such drugs are the NPS Pharmaceutical Company with a modulator for Ca sensors, and Xytis with a glycine site enhancer for NMDA channels under development (see homepages of NPS Pharmaceuticals Inc. and Xytis Inc.).
Of course, examples may appear where modulators are less beneficial than, for instance, weak partial agonists (Metha & Ticku 1999).
(c) Highly selective modulators for specific receptor subtypes are more likely to be found than subtype specific agonists, primary ligands, due to the evolutionary conservation of the primary binding sites for endogenous ligands (Birdsall et al. 1999; Spalding et al. 2002; Lanzafame et al. 2006; Langmead et al. 2006; Langmead & Christopoulos 2006). This is especially true for the muscarinic bindings sites. Actually, development of allosteric modulator drugs is an exploding field involving several effector molecules as indicated by recent publications listed in Table 7.2.
The receptor subtype selectivity by modulators, when judged to be based mainly on the conformational changes by induced fits or stabilization rather than on (high) affinity, is a kind of conformational change dubbed ‘absolute subtype selectivity’ (Lazareno et al. 2004).
During evolution, modulation may have been adapted for several secondary ligands with a more loose structure of their secondary binding sites. It has been difficult to find selectivity much above a factor 10 between primary ligands developed for muscarinic receptor subtypes (Alexander et al. 2006), whereas it seems likely that much higher selectivity may be found for modulators at the various muscarinic receptor subtypes (Birdsall & Lazareno 2005; Langmead et al. 2006). Other examples of allosteric modulators are for adenosine (Jacobson & Gao 2006; Childers et al. 2006), metabotropic glutamate (Kew 2004; Ritzen 2005; Shipe et al. 2005; Vauquelin & Liefde 2005; Vieira et al. 2005; Pin et al. 2005; Foster & Kemp 2006), and dopamine (Schetz 2005) receptor subtypes. Examples for ligand-gated ion channels (LGICs) are reviewed by Hogg et al. (2005) with benzodiazepines as the earliest prototype of modulators with allosteric responses at the GABA-A receptor (Olsen et al. 2004). For transporters, see, e.g., Maki & Dey (2006), Ghosh et al. (2006) and Borst et al. (2006), and other examples mentioned in sub-chapter 7.9 and Table 7.2.
Protocols and screens for new allosteric modulators for GPCRs are evolving (Langmead 2007; Raddatz et al. 2007; May et al. 2007; Avlani et al. 2007). Moreover, analytical tools are created to solve complex behavior in GPCR signaling (Fig. 7.31; Ehlert & Griffin 2008).
7.12.2. Summary of Modulator Benefits
A summary of the advantages of developed modulators compared with drug agonists is: (1) they mimic the function of endogenous primary ligands, (2) they are less likely to elicit adverse effects, and (3) they have possible greater receptor subtype selectivity than agonists and other molecules operating at a primary binding sites with its evolutionary conserved geometry.
Due to this rapidly expanding and potentially huge market for allosteric drugs, in the future, on theoretical grounds, it will be necessary to firmly implicate the ATSM and HOTSM for analysis of (allosteric) modulators in basic as well as applied approaches.
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Drug-acceptor interactions ISBN 978-91-977071-0-7 (print), 978-91-977071-1-4 (online)
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