A node in the cellular network is illustrated by only two populated states, active and inactive, separated by a sizeable but surmountable free energy barrier. Before activation, the inactive state dominates the population as indicated by the relative basin depth in the free energy landscape and a balance level. Within a narrow increment range in ligand concentration, the allosteric activation event shifts the population in favor of the active state. The activation is highlighted in the embedded plot with a typical sigmoid transition from the inactive to the active state.

The functional efficacy of a protein depends on how it populates its active conformation. As a node in the cellular circuit, a protein is expected to be able to switch its functional mode between on/off states. To fulfil its biological role, a protein has been optimized by evolution not only to populate a single active conformation; instead, two or more switchable states including the folded or disordered state need to be populated as well. Each conformation (or state) corresponds to a local free energy minimum at the bottom of the folding funnel. If the triggering event that effectively switches the protein population from one state to the other is far away from the active site of the protein, as for example in the case of substrate binding or covalent modification (PTM) distal to the functional site, it is referred to as allostery and the trigger site is termed an allosteric site. Since allostery plays significant roles in the cell, two distinct but compensatory approaches have been developed to explain how allostery works: one from a thermodynamic viewpoint, the other from a structural standpoint.

The thermodynamic view focuses on how to accurately describe allosteric phenomena, such as the classical cooperative oxygen binding to haemoglobin, without involving detailed structural information. The mathematical models below are mainly based on simple thermodynamic principles. On the other hand, the structural view emphasizes that optimized allosteric communication (or coupling) between allosteric and active sites is responsible for the allosteric conformational switch. It envisages that strain energy, created at an allosteric site by a triggering event, will propagate to the active site via an allosteric coupling channel to alter the functional conformation. The sequence-based statistical method for estimating thermodynamic coupling between residues in proteins has successfully visualized allosteric coupling in long-range energetic interactions for three proteins [44]. This concept has also inspired the discovery of allosteric binding sites and mutations via correlated movements from trajectories obtained from molecular dynamics simulations [55].

The simplest yet practical model for the thermodynamic view is the two-state model [36]. As defined in Figure 2A, a protein (or receptor) can populate one of two states: the inactive or active . In the free form, their distributions are governed by the equilibrium constant . According to the conformational selection concept, a ligand (A) will preferentially bind one state over the other. The binding affinities are quantitatively defined by association constants, with for an inactive and for an active conformation. The thermodynamic relationships in Figure 2A indicate that the binding affinity ratio of ligand for and , , becomes the allosteric intrinsic efficacy of ligand with . The population shift toward the active state following binding is proportional to the fraction of receptors in the activated state, . Manipulation of the ATSM equations given in Figure 2A works out the as a function of ligand concentration , with three independent parameters, , , and , giving . The basal activity is reduced to ; the maximum activity is given as ; and the activity at the middle point of the transition, , corresponds to a ligand concentration of . The summation of ATSM in Figure 2B shows the characteristic sigmoid activation curve from basal activity through a middle point transition to maximum activity.

The ATSM offers allosteric ligands a functional classification through the measured allosteric intrinsic efficacy, α. As indicated in Figure 4, a full agonist can reach nearly 100% activity with . A partial agonist corresponds to . A natural antagonist is defined by where the ligand shows no binding preference, in contrast to an inverse agonist where the ligand preferentially binds the inactive state conformation. The two conceptual allosteric frameworks, conformational selection and population shift, fuse through the same parameter α in the ATSM, however with differing descriptions, binding affinity, and allosteric intrinsic efficacy. Nonetheless, despite the simple yet powerful description, the ATSM is unable to provide a direct connection between structure and function. To link the thermodynamic view with the structural view, we next sketch the relative free energy between the two states in the ATSM.

The one-dimensional free energy landscape [10] provides a simple way to capture the relative populations of the conformations. As illustrated in Figure 3, the two local free energy minima occupied by the two conformations correspond to two populated states denoted inactive () and active (). In the absence of a ligand, is the most populated state (pale green); at a saturating ligand concentration, the state (orange) becomes the more populated state, indicating a population shift from to following ligand binding. Given the equilibrium constant, for the reaction , an exact measurement of the relative populations of P versus R can be expressed by Gibbs free energy , where R is the ideal gas constant and the temperature. In line with the ATSM, and in Figure 5A are the free energy differences between the and states, respectively for prior () and following () binding. Using the equilibrium equations defined in Figure 2A, we obtain and . When ligand A approaches its saturation concentration ( for or for ), the free energy difference, , can be approximated by . Then, the overall allosteric population shift due to ligand binding, expressed by the free energy change, is given by . In terms of the free energy landscape, the can be alternatively expressed by two components: , where refers to a favorable stabilization energy with respect to the active conformation through ligand binding, and to destabilization energy with respect to the inactive R conformation. In addition to Figure 5A with , to further clarify the relationship of , Figure S1 in the Supporting Information provides a free energy landscape with .

With a given L in the ATSM, the maximal extent of allosteric activation (Figure 2B) depends solely on the parameter . The extent of the population shift following binding, with a maximum free energy change of , also indicates dependence on . The population shift in a quantitative free energy landscape, already a simplified thermodynamic view, can also simplify the structural view of allostery, as shown in Figure 5B for two structural sites linked by an allosteric coupling constant . In terms of binding reversibility at both sites, the double arrow indicates that positive (or negative ) cooperative binding is bidirectional; binding at the functional site will increase (decrease) binding affinity at the allosteric site. The coupling constant determines the population shift of the active conformation due to binding at the allosteric site. The traditional structural view explains allostery by (1) the coupling between the two distal sites, which is described as strain energy created at the allosteric site, and releasing its energy through a propagation pathway toward the functional site with a consequent shift of the population; and (2) the highly correlated (or concerted) motion through a set of interacting residues which link the sites. Both explanations imply higher allosteric efficacy with a stronger coupling. However, the coupling through the propagation channel does not determine the allosteric efficacy; it is merely a necessary condition for allostery. What determines the allosteric efficacy is the sum of the extent of stabilization of the active conformation plus the destabilization of the inactive conformation following ligand binding.

Biased agonist binding implies that GPCRs can adopt multiple active states [56]. The structural view of the ATSM with one allosteric site and one functional site can be expanded to a system with one allosteric and two functional sites if the population of the two distinct active sites are regulated independently by the allosteric site. This argues that the two functional conformations are not mutually exclusive. Instead, they coexist and the total concentration of is independent for both active conformational states with . Figure S2 gives the equilibrium cycles for the two functional states with one ligand. The structural view with one allosteric site for two independent activities is given in Figure S3, showing two independent allosteric efficacies, α1 and α2, for two independent functional sites.

The discovery of allosteric modulators [35], [57]–[59] of GPCRs and even ternary synergistic allostery [60] requires an extension of the simplest allosteric two-state model to accommodate two or more allosteric ligands bound at distinct sites. Indeed, such an extension has been formulated by Hall [61]. The extended ATSM gives five more species including ligand , two binary complexes ( and for bound to inactive and active ), and two ternary complexes ( and ), with four additional parameters (, , , and , defined below). We refer to Hall's model as the extended ATSM with two allosteric ligands. Below, we only describe briefly the extended ATSM and refer the reader to the original paper [61]. In Figure 6A, ten species are related by four equilibrium cycles with seven parameters. In the first cycle (orange), the , , and in the extended ATSM are the same as for the single ligand ATSM, with L the equilibrium constant between the two states, the binding affinity of ligand bound to inactive , and the allosteric intrinsic efficacy of ligand . Similarly, the second cycle (pale green) defines and respectively as the binding affinity and the allosteric intrinsic efficacy of ligand . As revealed in the third cycle (cyan), provides the binding cooperativity between ligands and via formation of ternary complex . The parameter governs the cooperativity in the activation by ligands and through formation of as indicated in the fourth equilibrium cycle (red). The cubic shape of the cycles in the extended ATSM is given in Figure 6B.

As in the simplest ATSM, the free energy differences between the R and R* states in the extended ATSM are defined as and , respectively for prior ( and ) and following ([A]≠0 and [B]≠0) binding. Again, as both ligands approach their saturating concentrations, , ) can be approximated by . The population shift due to dual ligand binding as expressed by the free energy change becomes . The energy landscape of the extended ATSM is given in Figure S4. Overall, the free energy change for the population shift including the contribution from the efficacy of the second ligand is and the activation cooperativity . The corresponding structural view is in Figure 5C, with the allosteric efficacy reflecting independent communication between the two allosteric sites and the functional site. The communication between the two allosteric sites expressed by the binding cooperativity does not affect the allosteric efficacy. In contrast, the activation cooperativity provides the sum of the allosteric effects from the coupling between allosteric site 1 and the allosteric coupling plus from allosteric site 2 and allosteric coupling . As in the simpler ATSM, communications between the sites are through existing channels and ligand binding determines the allosteric efficacy and the activation cooperativity.

Two complex extensions of ATSM have been developed [62]. The discovery of biased allosteric modulators [31] of GPCRs provides a simple extension of the structural view, similar to the case of one allosteric site and two (independent) functional sites. If the populations of the two functional sites are regulated independently by two distinct allosteric sites, and the two functional conformations can coexist, the structural view can be described by two allosteric sites and two (independent) functional sites, as in Figure 6C. The allosteric coupling set (, , and ) for functional site 1 is added to a duplicated set of independent allosteric efficacies (, , and ) for functional site 2, with the shared binding cooperativity γ linking the two allosteric sites.

The simplified structural view of allostery with experimentally measureable parameters can usefully define agonist types. The classifications are summarized in Table 1. The system is specified in column 1, with the total number of allosteric and functional sites as defined by the structural view of allostery. A1 and A2 in column 1 correspond to the presence of one or two allosteric sites, and F1 and F2 to one or two functional sites. The first four classifications of the A1/F1 system (one allosteric and one functional site) are given in Figure 4 as full agonist, partial agonist, natural antagonist, and inverse agonist. In the rest of the classification, the four types of couplings between allosteric and functional sites are sorted into only two categories: agonist for and antagonist for . Two couplings in a system containing a single allosteric and two independent functional sites (A1/F2), α₁ and α₂, are in the Supporting Information (Figure S3). If both couplings are not in the same agonist category, it is classified as biased agonist. Otherwise it is classified as an agonist ( and ) or antagonist ( and ). In the A2/F1 system of Figure 5C, the positive () or negative () modulator is defined with , which means no direct coupling between the modulator and functional sites. Similarly, the A2/F2 system there can be defined as biased agonist with β₁ and β₂ not in the same agonist category and biased modulator with and , not in the same agonist category.

Above, we described allostery from three perspectives, thermodynamic, free energy landscape of population shift, and structural, with exactly the same allosteric efficacies emerging from the equations of the allosteric two-state model. Below, we explore how allostery works in each.

The thermodynamic view tells us that allostery works via a population shift from the inactive to the active state (), with the ligand preferring to bind the active over the inactive conformation. It is the preferred binding affinity, ( vs. ) not the overall binding affinity, that puts forth the allosteric effect. Although the binding preference () in the ATSM, which captures the conformation selection scheme (), fits the allosteric population shift (), the possibility of an activation path via induced-fit () cannot be ruled out. However, as long as the thermodynamic equilibrium is established rapidly, we do not concern ourselves with which path is preferred to reach the activation state. The free energy landscape of the ATSM also tells us that allostery works via population shift from the inactive to the active state with . This indicates that allostery can be quantified by allosteric efficacy () and confirms the implicit inference from the thermodynamic view. The more surprising implication results from expressing by two components, The first component , dubbed stabilization energy, specifies the amount of the active conformation being stabilized by the binding event; the second component , dubbed destabilization energy, corresponds the amount of the inactive conformation being destabilized through the ligand binding. When both the agonist-bound active structure and the inactive structure are known, these two energy terms can solve the fundamental underpinnings of allostery.

According to the structural view, the allosteric coupling between two sites does not involve ligand binding, which is the trigger of allostery. So, how does allosteric coupling between two sites help to explain allostery? First, coupling implies that the conformation at the allosteric site must to some degree be correlated with the conformation at the functional site. When the conformations at the functional site breathe dynamically between the two free energy basins of inactive and active states, conformational changes take place at the allosteric site. Even slight structural differences at the allosteric site can be sufficient to set the stage for a binding event, via correlated modulation of the population of the activated state. Second, the coupling also implies that certain key interactions are responsible for structural stability in only one state, active or inactive, but not both states. These shape the conformational change at the allosteric site.

Finally, we note that the implications of our unified model are mostly consistent with those of the recent EAM model [14], [17]; however, they differ in the key conceptual guideline for determining agonism. Our model emphasizes that it is the specific interactions between the ligand and the host protein, dubbed as the stabilization of the active state and destabilization of the inactive state, that determine the allosteric efficacy and agonism in a ligand binding event. In contrast, the EAM states that the allosteric mechanism is robustly encoded in the ensemble, and does not require different interactions for a ligand acting as an agonist versus as an antagonist [17]. The conclusion that ligand-specific interactions can be disregarded is not surprising, since the EAM model a priori assumes implicit fixed interactions between the ligand and the host. However, the energy terms defined in the EAM can be reformulated within the framework of the ATSM [36] to express allosteric efficacy; that is, the EAM can be considered as a yet another, different formulation of the ATSM. Thus, we believe that the implications of the EAM may have stemmed from the underlying premise of overlooking changes in the interactions between the ligand—agonist or antagonist—and the host, which in the EAM model were implicitly expressed by the distinct set of energies.

Our purpose in writing this Perspective is to survey points of view on allostery and synthesize them via a mathematical model to obtain a coherent understanding of the question of how allostery works. Starting with the allosteric two-state model, we link the thermodynamic model of allostery and the free energy landscape of population shift. This not only unifies allosteric models based on the same allosteric descriptors; it also provides a simplified structural view of allostery and a set of measureable parameters which allow distinguishing among agonist, biased agonist, modulator, and biased modulator. The emerging new unified view accommodates the three basic elements of allostery: the thermodynamics point of view, the free energy landscape of the population shift, and the structural point of view.

The thermodynamic view of the allosteric two-state model provides experimentally measurable parameters, which emphasize that allostery reflects preferred ligand binding to one of the two (active, inactive) states. The free energy landscape representation incorporates the preferred ligand binding formulation and transforms the conceptual framework of population shift into an amount of energy which is proportional to the stabilization of the active conformation and (or) destabilization of the inactive conformation, in the case of allosteric activation. The integrated view translates these into an important rule for the classical structural view of allostery: by itself, the structural coupling (or the propagation pathway) between the functional (active) and the allosteric sites does not involve allostery; it merely identifies high correlation between the conformations of the two sites. The importance of this rule can be assessed by the number of publications using couplings and propagation pathways to delineate allostery.

Beyond generalities, an important question is how does allostery work in a specific system of interest? To decipher the allosteric mechanism in a given protein one needs three elements: first, the active conformation which is responsible for the specific function. Structures are essential in order to understand how the agonists act as allosteric triggers. This conformation should populate one of the local free energy basins (the “active basin”) at the bottom of the folding funnel. This emphasizes that an allosteric event such as ligand binding does not create a new conformational state; it only shifts the population among existing states. Second, one needs a set of interacting residues which make up the allosteric communication pathway between the active and allosteric sites; this implies that evolution has set the relative populations of the active versus inactive conformations and linked them to critical pathway residues. Mutations of those residues—if not involving both interacting partners—will affect the allosteric propagation and the population of the active conformation. Knowledge of these mutations is useful because they can be explored by methods such as fast ensemble sampling [46] or covariance analysis of NMR chemical shifts [53]. Third, to figure out an activation event, we suggest considering whether ligand binding at the allosteric site stabilizes the active conformation or destabilizes the inactive conformation or both at the same time [1]. Collectively, we expect that the unified view of “how allostery works” will help guide allosteric drug discovery and provide structural insight into multiple signaling pathways mediated by biased agonists. Finally, the unified view helps unravel the structural mechanism of allostery and advances the notion that allosteric switches in cellular circuits, which are governed by optimized active and inactive conformations with dominant populations, co-evolved with their associated allosteric ligands.