Allosteric binding cooperativity in a kinetic context

Feature Allosteric binding cooperativity in a kinetic context Óscar Díaz 1,2,3 , Victor Martín 1,4 , Pedro Renault 1,2,3 , David Romero 5 , Antoni Guillamon 4,5 , Jesús Giraldo 1,2,3, ⇑ Allosteric modulators are of prime interest in drug discovery. These drugs regulate the binding and function of endogenous ligands, with some advantages over orthosteric ligands. A typical pharmacological parameter in allosteric modulation is binding cooperativity. This property can yield unexpected but illuminating results when decomposed into its kinetic parameters. Using two reference models (the allosteric ternary complex receptor model and a heterodimer receptor model), a relationship has been derived for the cooperativity rate constant parameters. This relationship allows many combinations of the cooperativity kinetic parameters for a single binding cooperativity value obtained under equilibrium conditions. This assessment may help understand striking experimental results involving allosteric modulation and suggest further investigations in the field. Keywords: allosteric modulation; binding kinetics; binding cooperativity; rate constant; cooperativity rate constant; residence time; GPCRs; heterodimer receptor Allosteric modulation at equilibrium conditions Allostery, in particular in G-proteincoupled receptors (GPCRs), is a research area of special interest to both academia and the pharmaceutical industry because of the known advantages (a ceiling effect level and greater GPCR subtype selectivity, among others) that allosteric modulators have with respect to orthosteric ligands. 1 In this study, we consider first the allosteric ternary complex receptor model, in which a receptor R bears two binding sites to which the orthosteric ligand A and the allosteric ligand B bind. 2,3 In the first instance, we do not consider how ligand binding translates into receptor function and we limit the analysis to a pure binding scenario in which either equilibrium or rate constants are used. At equilibrium, the concentrations of the four receptor species present in the system are regulated by the corresponding K 1 to K 4 equilibrium dissociation constants (Fig. 1a). 2,3 We can change the notation and introduce the aand bbinding cooperativity parameters (Fig. 1b). ameasures the binding affinity of A to RB with respect to the free receptor R, whereas bmeasures the binding affinity of B to AR with respect to R. Only three of the four equilibrium constants are independent or, in other words, ais equal to b. From now on, we use ato denote the binding cooperativity parameter between the two ligands. acan be greater, lower, or equal to one, indicating, respectively, an increase, decrease or no effect on the affinity of each of the ligands for the receptor because of the presence of the other bound ligand. Binding cooperativities can be experimentally measured by different methods and allosteric modulators can be classified as positive, negative, or neutral depending on a>1, a<1,ora= 1, respectively. 2 However, the mutual influence between the two ligands can be further examined when time is considered. For example, we can FEATUREFEATURE Drug Discovery Today dVolume xxx, Number xx dxxxx 2022 PERSPECTIVE FEATURE 1359-6446/Ó2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1016/j.drudis.2022.103441 www.drugdiscoverytoday.com 1 Drug Discovery Today d Volume 28, Number 2 d February 2023 FEATURE ask whether it is possible to find a positive allosteric modulator (PAM) B that is more kinetically unstable in its binding site when A is present than when it is bound to the free receptor. If a> 1, one would expect an increase in the affinity of each of the two ligands when the other is present, but this conclusion does not explain the behavior of the ligands and the mutual influence between them when they are already at the receptor binding site and only the processes of ligand–receptor dissociation are considered. From equilibrium constants to rate constants: including the kinetic context Equilibrium constants are the ratio between rate constants (Fig. 2a). The inclusion of rate constants in the analysis opens the discussion to binding kinetics, a pharmacological research area of major application in clinical and drug discovery research. 4,5 We follow the same rationale as in Fig. 1 but using rate constants (association: k +1 to k +4 and dissociation: k 1 to k 4 )(Fig. 2a) and their corresponding a + , a  ,b + , and b  cooperativity rate constant parameters, with a + =k +4 /k +1 ,a  =k 4 / k 1 ,b + =k +3 /k +2 , and b  =k 3 /k 2 (Fig. 2b). It can be shown that only seven of the eight rate constants are independent: (k 1 k +4 )/(k +1 k 4 )=(k 2 k +3 )/(k +2 k 3 ), or, in other words, a + /a  =b + /b  . In the same way as equilibrium constants are the ratio between rate constants, the binding cooperativity parameters (a and b) are the ratio between the corresponding cooperativity rate constant parameters (a=a + /a  and b=b + /b  ). Given a=b, this does not necessarily mean that a + =b + and a  =b  . The latter is a sufficient but not a necessary condition and, thus, the kinetic reciprocity between orthosteric and allosteric ligands occurs at the level of their cooperativity rate constant ratios and not their absolute values. Let us consider the following example: a=b= 4, with a + =2,a  = 0.5, b + =8 and b  = 2 for an orthosteric ligand A and an allosteric ligand B. These values imply a PAM in binding terms (a>1) and, thus, the equilibrium dissociation constants of both the orthosteric and the allosteric ligands decrease in the presence of the other bound ligand or, in other words, their affinities for the receptor increase in the presence of the other Drug Discovery Today FIG. 2 Allosteric ternary complex model. It is assumed that A is the agonist and B is the allosteric modulator. (a) The equilibrium dissociation constants in Fig. 1 in the main text are substituted by the corresponding association (k +1 to k +4 ) and dissociation (k 1 to k 4 ) rate constants: K 1 =k 1 /k +1 ,K 2 =k 2 /k +2 ,K 3 =k 3 /k +3 , K 4 =k 4 /k +4 .(b) k +3 ,k 3 ,k +4 , and k 4 have been removed by including a + ,a  ,b + , and b  cooperativity rate constant parameters, with a + =k +4 /k +1 ,a  =k 4 / k 1 ,b + =k +3 /k +2 , and b  =k 3 /k 2 . It can be shown that only seven of the eight k +1 to k +4 and k 1 to k 4 rate constants are independent: if we express each equilibrium dissociation constant in terms of concentrations of receptor species, then K 1 /K 4 =K 2 /K 3 = [R][ARB]/([AR][RB]) (Fig. 1 in the main text) and, by putting equilibrium constants in terms of rate constants, it can be seen that (k 1 k +4 )/(k +1 k 4 )=(k 2 k +3 )/(k +2 k 3 ) (one rate constant depends on the other seven), or, in other words, a + /a  =b + /b  . Adapted from 3 . FEATURE Drug Discovery Today FIG. 1 Allosteric ternary complex model. It is assumed that A is the agonist and B is the allosteric modulator. (a) K 1 to K 4 equilibrium dissociation constants are used: K 1 = [A][R]/[AR]; K 2 = [B][R]/[RB]; K 3 = [B][AR]/[ARB]; K 4 = [A][RB]/[ARB]. (b) K 3 and K 4 have been removed by including aand bbinding cooperativity parameters, with a=K 1 /K 4 and b=K 2 /K 3 . It can be shown that only three of the four K 1 to K 4 constants are independent: if we substitute each equilibrium dissociation constant by its expression in terms of concentrations of receptor species, then K 1 /K 4 =K 2 /K 3 = [R][ARB]/([AR][RB]) (one equilibrium constant depends on the other three), or, in other words, a=b. Adapted from 2 . FEATURE Drug Discovery Today d Volume 28, Number 2 d February 2023 2www.drugdiscoverytoday.com bound ligand. However, if we look at the a  and b  parameter values, we see opposite effects. Whereas the dissociation rate constant of the orthosteric ligand decreases when the allosteric ligand is bound (a  = 0.5), that of the allosteric ligand increases when the orthosteric ligand is bound (b  = 2). Moreover, if we look at the a + and b + parameter values, we see that both are > 1 although the effect of the orthosteric ligand on the association rate constant of the allosteric modulator (b + = 8) is greater than that of the allosteric modulator on the association rate constant of the orthosteric ligand (a + = 2). Interestingly, for this particular a=b=4 value, many other combinations of values of the cooperativity rate constant parameters are possible, which indicates that different mechanistic hypothesis concerning microscopic events are compatible with a macroscopic outcome. For instance, the a=b= 4 binding cooperativity value can also result from a + =8,a  =2,b + = 0.8 and b  = 0.2. Now, the effects that the allosteric modulator and the orthosteric ligand exert on each other are opposite in both the association and the dissociation rate constants (a + >1 and b + <1; a  >1 and b  <1). See the discussion below on the relationship between residence time and agonist efficacy and 6 for an insightful review on the influence of allosteric modulators on the binding kinetics of the orthosteric ligand. In this report 6 , adenosine receptors were selected for analysis and, among others, adenosine A 3 allosteric modulators were examined. For purposes of illustration, two compounds are now taken. The first, VUF5455, behaved as an A 3 PAM by significantly retarding the dissociation rate of the agonist radioligand [ 125 I]-I-ABMECA, from the adenosine A 3 receptor in a concentration-dependent manner. 7 Interestingly, its effect on the dissociation rate of the antagonist [ 3 H]-PSB-11 was insignificant. 6,7 These data illustrate the known dependence of allosteric modulation on the orthosteric ligand used, which is reflected in both equilibrium and kinetic assays. 1,8 The second compound, LUF6096, which bears a different chemical scaffold, also behaved as a PAM of the adenosine A 3 receptor. 9 Noteworthy, the compound was able to change the biphasic dissociation of [ 125 I]-I-AB-MECA from the receptor into a monophasic process, by slowing the kinetics of the agonist in the fast dissociating phase (k off_fast = 0.089 to 0.035 min 1 ). This effect was attributed to the stabilization of the receptor active conformation. This proposal was corroborated by a functional assay, in which LUF6096 significantly enhanced the intrinsic activity of ClIBMECA agonist. 6,9 These are two examples showing the effects of allosteric modulators on the dissociation rate constants of orthosteric ligands. As recognized in 6 , the influence of allosteric modulators on association rate constants of orthosteric compounds has been less investigated, which indicates that further work is needed to cover the entire kinetic space. However, the occurrence of receptor–receptor interactions in GPCRs adds a layer of complexity to the concept of allosterism. Allostery in a heteromeric context Allostery can arise not only from the interaction between an orthosteric and an allosteric ligand within a single receptor protein, but also from the interaction between two or more orthosteric ligands in an oligomeric receptor. Given its analogy with the previous case, at least at the mathematical level, we consider the model of receptor heterodimerization. Previously, 10 we proposed a mathematical model for receptor heterodimerization. Fig. 3a shows an adaptation of the binding part of the model in which kinetic constants for the binding of ligands A and B to the corresponding R 1 and R 2 receptor protomers in the heterodimer are included. Figs. 2 and 3 depict two models describing allostery between two ligands. Although the models are different in terms of protein structure (one or two proteins, respectively), they are equivalent to each other from the point of view of the kinetic parameters involved. This means that, in the absence of structural information about the allosteric interactions between the two bound receptor–ligands, binding experimental data are compatible with both a monomeric and a heterodimeric receptor. Therefore, the discussion above on the variability of cooperativity kinetic parameter values within a common a + / a  =b + /b  ratio is also valid. At this point, it is worth comparing heterodimeric receptors with homodimeric receptors. We see that a homodimeric receptor is a particular case of a heterodimeric receptor with R 1 =R 2 = R or, in other words, a homodimeric receptor is a limitDrug Discovery Today FIG. 3 Heterodimer receptor model considering only the binding part. A is a ligand selective for R 1 and B is a ligand selective for R 2 .(a) k +1 to k +4 association and k 1 to k 4 dissociation rate constants are used. (b) k +3 ,k 3 ,k +4 and k 4 have been removed by including a + ,a  ,b + , and b  cooperativity rate constant parameters, with a + =k +4 /k +1 ,a  =k 4 /k 1 ,b + =k +3 /k +2 , and b  =k 3 /k 2 . Proceeding analogously as in Fig. 2 in the main text, it can be shown that only seven of the eight k +1 to k +4 and k 1 to k 4 rate constants are independent, (k 1 k +4 )/(k +1 k 4 )=(k 2 k +3 )/(k +2 k 3 ), or, in other words, a + /a  =b + /b  . Adapted from 10 . FEATURE Drug Discovery Today d Volume 28, Number 2 d February 2023 FEATURE www.drugdiscoverytoday.com 3 ing case of a heterodimeric receptor in which the two protomers progressively resemble each other until finally they are the same. 11 If, in addition, only one ligand species (say A) is included, then the previous relationship between cooperativity rate constants is simplified (a + =b + and a  =b  ) and only two cooperativity rate constants (a + and a  ) contribute to the homodimer receptor system 12 (Fig. 4). In functional terms, a heterodimer receptor with two orthosteric sites is more complex than a monomeric receptor with an orthosteric site and an allosteric site. In the heterodimer, at least two signaling pathways are present, one for each protomer, leading to a more complex functional scenario. The present study is mainly aimed at binding. However, it appears clear that the mutual influence in binding kinetics that ligands can have on each other could affect their respective functional responses. The potential variability in cooperativity kinetic parameter values might be obscured under equilibrium conditions but can be determinant when equilibrium is not present. In this regard, it can be interesting to consider whether complex pharmacological problems, such as that described in 13 can be reanalyzed through binding kinetics. In 13 , the potential relationship of the 5HT2A-mGlu2 heteromer with schizophrenia was postulated. For this heteromer, a Gq signaling pathway is linked to the 5HT2A protomer, whereas a Gi signaling pathway is linked to the mGlu2 protomer. In healthy circumstances, a determined Gi-Gq balance is present, which is disrupted by a decrease in Gi and an increase in Gq signaling under schizophrenia conditions. The authors found that, in general, dominant (strong) agonists enhance signaling through the protomer they target as part of the heteromer but inhibit signaling of the heteromeric receptor partner. By contrast, inverse agonists inhibit signaling through the protomer they target as part of the heteromer but enhance signaling of the heteromeric receptor partner. 13 Thus, to restore normal balance in patients with schizophrenia, an mGlu2 dominant agonist would be appropriate to increase Gi signaling and additionally decrease Gq signaling. In the same way, a serotonin 5HT2A inverse agonist would be appropriate to decrease Gq signaling and additionally increase Gi signaling. Furthermore, a combination of the two ligands would synergistically favor the desired effect. 13 This behavior was modeled in 14 by using a heterodimer model under equilibrium conditions. 10 To do so, proper values for the parameters describing receptor function under equilibrium conditions were chosen; in particular, values either greater than one or lower than one for the functional cooperativities in their respective Gi or Gq signaling pathways were chosen. 14 A second layer of complexity comes from considering the mutual influence between the receptors through ligand binding. Thus, if we are including a combination of two ligands, that is, a strong agonist A for mGlu2 and an inverse agonist B for 5HT2A, a binding cooperativity a>1 would favor the binding of the two ligands. Interestingly, and as discussed above, a single avalue can be obtained from different (a + ,a  ) and (b + ,b  ) combinations. This variability in association and dissociation cooperativity rate constants can yield striking and unexpected results for those cases not restricted to equilibrium conditions, which could provide new insights into the biological problem. Furthermore, this mechanistic knowledge can help design the appropriate protocol for combination drug therapy in neurologic and psychiatric diseases. 15 Residence time, agonist efficacy, and allosteric interactions Binding kinetics and, consequently, the time factor are conceptual pieces in the mechanism of drug action that should not be neglected in pharmacological research and pharmaceutical development. The time factor is present in an explicit way through the concept of residence time. 16 If we define residence time as the time a ligand spends at the receptor-binding site, many different combinations of cooperativity rate constant values can be obtained from different ligands with similar equilibrium dissociation constants. In the above proposed case (a=b= 4, with a + =2,a  = 0.5, b + = 8, and b  = 2), if no other factors are considered, there would be expected an increase in the residence time of ligand A because of the presence of ligand B (a  <1) and a decrease in the residence time of ligand B because of the presence of ligand A (b  >1). There are studies in the literature showing the effect of allosteric modulators on the binding kinetics of orthosteric ligands. As examples, we can mention values included in Table 7 of 6 showing the Drug Discovery Today FIG. 4 Homodimer receptor model constructed from the heterodimer model shown in Fig. 3 in the main text by considering R 1 =R 2 = R and A = B. (a) The set of rate constants present in Fig. 3a in the main text is reduced because k +2 =k +1 ,k 2 =k 1 ,k +4 =k +3 and k 4 =k 3 .(b) k +3 and k 3 have been removed by including a + and a  cooperativity rate constant parameters, with a + =k +3 /k +1 and a  =k 3 /k 1 . Comparing Fig. 3b in the main text with (b), it can be seen that k +2 =k +1 ,k 2 =k 1 ,a + =b + and a  =b  . FEATURE FEATURE Drug Discovery Today d Volume 28, Number 2 d February 2023 4www.drugdiscoverytoday.com decrease in the dissociation rate constant of the adenosine A 3 [ 125 I]-I-AB-MECA agonist exerted by some allosteric modulators, namely, 43% (VUF5455), 7 46% (DU124183), 17 58% (2-AG), 18 and 47% (HMA). 19 Interestingly, the HMA allosteric modulator increased 1.6-fold the dissociation rate constant of the [ 3 H]-PSB-11 antagonist. 19 Residence time is expected to be positively correlated with agonist efficacy because the longer an agonist remains bound to the receptor, the more cycles of G-protein activation it can catalyze. 8 This proposal has been proven in some receptor systems, such as the M 3 muscarinic acetylcholine, 20 the adenosine A 2A , 21 the adenosine A 3 , 9 and the b 2 adrenergic 22 receptors, but not in others, such as the adenosine A 123 and dopamine D 224 receptors (see 8 for a discussion). These discrepancies can be a consequence of the intrinsic complexity of the relationship between efficacy and residence time when, for example, allosteric effects coming from lipid–receptor 25,26 or receptor–receptor 11,27 interactions might be present (reviewed in 8 ). These interactions might differently modulate agonist-binding kinetics, yielding a functional result that is the product of a combination of association and dissociation rate constants. In this regard, dissociation rate constants determine the time ligands spend in the receptor binding site. Yet, before dissociating from the receptor, the ligand must bind to it. Thus, dissociation and association rate constants influence drug action and should be considered together. 28,29 On the one hand, high association rate constants can be useful in pharmacological therapy to allow fast association of the drug to its target. 30,31 On the other hand, although dissociation rate constants might be fundamental for drug action, this is not always the case because, as found in 32 , the prolongation of binding owing to a long drug–target residence time can only occur when the binding dissociation is slower than the pharmacokinetics (PK) elimination. PK is beyond the scope of the present report, which is limited to the study of cooperativity from a binding kinetics perspective and applied to two receptor models: a ternary complex receptor model and a heterodimer model. Moreover and to make the picture more complex, when relating residence time with efficacy, the former should include only the time the ligand spends bound to active receptor conformations, such as, for instance, the results in 9 and reviewed in 6 , in which the adenosine A 3 receptor allosteric compound LUF6096 specifically stabilized the active conformation of the receptor with a concomitant increase in the intrinsic efficacy of orthosteric agonist Cl-IBMECA. In this regard, we have to take into account that the receptor species included in Figs. 1–4represent macroscopic terms, including populations of different receptor conformations and states. Thus, inactive and active receptor species, either free or ligand bound, are present in the system. Moreover, if we attribute the observed receptor effect to receptor–G protein interactions, G protein-bound receptors are also implicitly included in the receptor terminology. The question arises on how the schemes in Figs. 1–4represent this molecular variety. For the sake of simplicity, we denote R as the free receptor and LR as the ligand-bound receptor in the figures. When using mathematical models of receptor function, it is said that a stimulus is provided by each of the receptor species through the product of the concentration of the considered receptor species and the corresponding intrinsic efficacy e(S R =e R [R] and S LR =e LR [LR]); where, if L is an agonist then e LR >e R ; if L is an inverse agonist then e LR <e R ; and if L is a neutral antagonist e LR =e R . Then, these stimuli are summed up (S = S R +S LR ) and converted into effect through the transducer function E = E m S/ (S + K E ), with E m the maximum possible effect and K E the transducer parameter. 33 Yet, and speaking in molecular terms, for the receptor to generate a stimulus, it is necessary that an active conformation is formed. If R and LR now denote inactive receptor conformations, R* and LR* are the corresponding active ones. If, in addition, the G protein is the transducer protein involved in the signaling pathway, we can accept that R*G and LR*G represent the receptor species responsible for the produced stimuli and, subsequently, for the observed effect. We can consider the binding kinetics concept through the association and dissociation rate constants of the different species of the system and, most importantly, of those related with the generated stimuli. Thus, we can consider that a low LR* dissociation rate constant (high residence time of the ligand in the active receptor complex) would be beneficial to allow the binding of the G protein. By contrast, if we focus our attention on the free receptor, initial receptor stimulus through R*G precoupling would be increased by an agonist with a high association rate constant for this receptor–G protein complex, because the intrinsic efficacy of an agonist–receptor complex (e LR ) is higher than that of the free receptor (e R ) (see 34,35 for detailed descriptions of GPCR kinetics). These are two examples of microscopic events showing how either decreasing or increasing dissociation or association ligand–receptor rate constants, respectively, can both increase the efficacy of the system. These changes in ligand–receptor rate constants both will lead to a decrease in the ligand–receptor equilibrium dissociation constant. The increase in efficacy will result only if a concomitant increase in the efficiency of G protein activation is part of the process. To quantitatively illustrate these concepts, a simulation of the biological response under the heterodimer receptor model depicted in Fig. 3 was performed. 10 To this end, the transducer function E/E m = S/(K E + S) proposed above was used, with the total stimulus S defined as S = e [R 1 R 2 ]+e A [AR 1 R 2 ]+e B [R 1 R 2 B]+e AB [AR 1 R 2 B]. The model has the complexity of including two ligands, A and B, which are selective for protomers R 1 and R 2 , respectively. We assume that both ligands are in excess with respect to the total receptor concentration. The model includes constitutive receptor activity through the eparameter and ligands A and B are agonists, neutral antagonists, or inverse agonists depending on the values of their intrinsic efficacies, e A and e B , compared with e. The doubly bound receptor has an intrinsic efficacy, e AB ,defined as e A e B d. In a similar way to the binding cooperativity a,dcan be greater than, lower than, or equal to one, thus reflecting the mutual allosteric interaction between the two ligands at the functional level (see 10 for a detailed description of the heterodimer model). For the sake of simplicity, we examined the biological effect of changing the concentration of various agonists in the presence of a constant concentration of allosteric modulators (Fig. 5). To analyze the effect of binding kinetics on the transducer function, some parameters were FEATURE Drug Discovery Today d Volume 28, Number 2 d February 2023 FEATURE www.drugdiscoverytoday.com 5 changed with respect to a reference condition (Fig. 5, black curve). First, we considered the effects of either decreasing the dissociation rate constant of ligand A through k 1 and k 4 rate constants or increasing the association rate constant of ligand A through k +1 and k +4 rate constants (Fig. 5, red curve). The values used were chosen so that the (k 1 k +4 )/(k +1 k 4 )=(k 2 k +3 )/(k +2 k 3 ), or, in other words, a + /a  =b + /b  condition was satisfied. We can see that a change in binding kinetics has an effect only on the observed potency of ligand A (shift to the left with respect to the reference curve) with no change in efficacy (maximum response). However, as discussed above, a change in the asymptotic maximum effect can be observed if the change in binding involves a change in intrinsic efficacy. To illustrate this, the blue curve in Fig. 5 was obtained from the red curve by assuming an increase in e A that might result from either a decrease of the dissociation rate constants or an increase of the association rate constants of ligand A. Both changes might increase the biological response: in the former case, by assuming active AR 1 R 2 and AR 1 R 2 B receptor states in which a low dissociation rate constant of ligand A favors the binding of the G protein; in the latter case, by assuming active R 1 R 2 and R 1 R 2 B receptors coupled to G proteins in which ligand A presents a high association rate constant for the complexes. The maximum response displayed by the agonist A can also be increased through the mutual allosteric effect between ligands A and B. To show this point, the green curve in Fig. 5 was obtained from the red curve by assuming an increase in the dparameter from 5 to 100. This increase in dleads to an increase in e AB and, as a result, an increase in the concentrations of active AR 1 R 2 B receptor states. To generalize the concept of allosterism, we draw attention again to the conceptual similarities between the allosteric ternary complex receptor model and the heterodimer receptor model, where dis present in both (see 36 for a discussion on operational models of allosterism). As mentioned above, some, but not all, experimental studies have found a correlation between residence time and efficacy. In particular, we can recall on a study on the adenosine A 2A receptor, 21 in which the authors found a correlation between the residence time of an agonist and its functional efficacy in two assays; they also found that, compared with the equilibrium affinity, the receptor residence time of the A 2 receptor agonist had a much betDrug Discovery Today FIG. 5 Simulation of the E/E m fractional effect resulting from the heterodimer binding kinetics model depicted in Fig. 3 in the main text. The translation of binding into function is made through the relationship E/E m = S/(K E + S), with the total stimulus S defined as S = e[R 1 R 2 ]+e A [AR 1 R 2 ]+e B [R 1 R 2 B] + e AB [AR 1 R 2 B], where e, e A ,e B , and e AB =de A e B are the intrinsic efficacies of the free receptor, the singly bound A and B receptors, and the doubly bound receptor, respectively; d measures the functional interaction between A and B, and K E is the transduction factor of stimulus into effect. 10 The reference curve is the black curve, which includes the following functional parameter values: w=[R T ]/K E =1,e=1,e A = 10, e B =10 –1 , and d= 5 and the following binding kinetics parameter values: k +1 =10 7 ,k 1 =10 –1 ,k +2 =10 7 ,k 2 =10 –2 ,k +4 = 2*10 7 ,k 4 =10 –3 ,k +3 = 4*10 7 ,k 3 = 2*10 –4 , with M 1 s 1 and s 1 units for association and dissociation rate constants, respectively. Ligands A and B are an agonist and an inverse agonist, respectively, because their intrinsic efficacies are greater and lower than that of the free receptor, e, respectively. Using cooperativity rate constant parameters, it can be seen that a + =k +4 /k +1 =2,a  =k 4 /k 1 =10 –2 ,b + =k +3 /k +2 =4 and b  =k 3 /k 2 = 2*10 –2 . Moreover, it is found that a + /a  =b + /b  = 200. The allosteric compound B is present at 10 –6 M fixed concentration. Red curve: the dissociation rate constants k 4 and k 1 are decreased with respect to the black curve, that is, k 4 =10 –5 and k 1 =10 –3 . The values for the cooperativity rate constant parameters are the same. The decrease in the k 4 and k 1 dissociation rate constants translates into a decrease in the dissociation equilibrium constant of A for R 1 and, consequently, into an increase in the potency of A, which is reflected in the displacement of the red curve to the left with respect to the black one. Note: the red curve can be equally obtained by an increase in the association rate constants k +4 and k +1 , with respect to the black curve, that is, k +4 = 2*10 9 and k +1 =10 9 . Blue curve: taking the black curve as a reference, k 4 and k 1 are changed in the same way as for the red curve, that is, k 4 =10 –5 and k 1 =10 –3 , but, in addition, there is an increase in the intrinsic efficacy associated with ligand A, e A = 50. As a result, there is an increase in the potency and efficacy of the system with a displacement of the black curve both left and upwards (the maximum response is increased). Green curve: taking the black curve as a reference, k 4 and k 1 are changed in the same way as in the red curve, that is, k 4 =10 –5 and k 1 =10 –3 , but, in addition, dis changed from 5 to 100. As a result, there is an increase in the potency and efficacy of the system with a displacement of the black curve both left and upwards (the maximum response is increased). Note: mathematical expressions of the maximum response (the limiting value of E/E m as [A] or [B] increase) can be found in 14 . FEATURE FEATURE Drug Discovery Today d Volume 28, Number 2 d February 2023 6www.drugdiscoverytoday.com ter correlation to its intrinsic efficacy. Similarly, in a study of the M 3 muscarinic acetylcholine receptor involving seven agonists, 20 the authors did not find a relationship between agonist efficacy and the equilibrium binding affinity. However, when efficacy was compared with the dissociation rate constant, a high correlation was found, suggesting a relationship between the duration of agonist binding at the receptor and the intrinsic efficacy. 20 Apart from experimental studies, a typical field in which the discussion on residence time makes particular sense is molecular dynamics (MD) simulations. In the case of a PAM in the context of binding to the receptor (a> 1), one would expect that, in those MD simulations including both the orthosteric and the allosteric ligands, the stability of ligand–receptor interactions of both ligands in their binding sites would increase compared with their simulations in the absence of the partner ligand. However, if this were not the case, we can now understand, based on the above discussion on a  and b  values, that a variety of results can be obtained from the dynamic interactions between orthosteric and allosteric ligands that can be compatible with a single cooperativity binding parameter obtained at equilibrium conditions. Concluding remarks Inclusion of cooperativity in a binding kinetic model of allosterism has enabled us to find a mathematical expression (a + / a  =b + /b  ) that links the cooperativity rate constants of the orthosteric and allosteric ligands. The expression shows that many different combinations of kinetic allosteric effects are possible for a particular value of the abinding cooperativity parameter obtained under equilibrium conditions. This assessment could help understand striking experimental results involving allosteric modulation and suggest further investigations in the field. Furthermore, the fact that allosteric modulators can exert pathway-specific effects leads to the concept of biased allosteric modulation, 37 a chemical space in which kinetically oriented drug discovery programs can help in the search for new pharmacological therapies. Data availability Data will be made available on request. Acknowledgments This project received funding from the European Union’s Horizon2020 research and innovation programme under grant agreement No 848068 and by the grant PID2020-119136RB-I00 funded by MCIN/ AEI/10.13039/501100011033. 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Óscar Díaz 1,2,3 , Victor Martín 1,4 , Pedro Renault 1,2,3 , David Romero 5 , Antoni Guillamon 4,5 , Jesús Giraldo 1,2,3, ⇑ 1 Laboratory of Molecular Neuropharmacology and Bioinformatics, Unitat de Bioestadística and Institut de Neurociències, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain 2 Instituto de Salud Carlos III, Centro de Investigación Biomédica en Red de Salud Mental, CIBERSAM, Spain 3 Unitat de Neurociència Traslacional, Parc Taulí Hospital Universitari, Institut d’Investigació i Innovació Parc Taulí (I3PT), Institut de Neurociències, Universitat Autònoma de Barcelona, Spain 4 Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain 5 Centre de Recerca Matemàtica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain ⇑ Corresponding author: FEATURE FEATURE Drug Discovery Today d Volume 28, Number 2 d February 2023 8www.drugdiscoverytoday.com