A cohesive model framework of receptor pharmacology: beyond the Emax model
Xiao Zhu (1), David B Finlay (2), Michelle Glass (2), Stephen B Duffull (1)
(1) Otago Pharmacometrics Group, School of Pharmacy, University of Otago, Dunedin, New Zealand; (2) Department of Pharmacology and Toxicology, University of Otago, Dunedin, New Zealand
Background: Just over 150 years ago, the Norwegian mathematician Cato Guldberg and the chemist Peter Waage, propounded the law of mass action [1]. A.V.Hill was the first to apply this mathematic principle to physiology [2]. In his studies on nicotine and curari, on the basis of the law of mass action and mass balance, he derived what was later known as “Emax model”. In pharmacological nomenclature, it is conventionally written as: E = Emax*A/(KA+A), where A is the concentration of a ligand and KA the equilibrium binding of the agonist. The Emax model is probably the most widely used model to describe drug-receptor interactions whether at the level of binding or a bioassay of a response variable. By linking pharmacokinetics (PK) to pharmacodynamics (PD), the Emax model (now driven by the concentration-time profile) provides a practical tool to describe concentration-effect relationship [3]. However, Emax model is so ingrained in PD modelling that the assumptions attached to it are often overlooked. This may cause difficulties in the interpretation of estimated parameters and extrapolation of the findings across different contexts (e.g., cell, tissue, animal and human). Hence, it is useful to have a framework that underpins our understanding and use of the Emax model.
Objectives: The overall goal of this work is to develop a cohesive model framework for the Emax model that allows generalisation of its application to meet a diverse range of experimental conditions. This encompasses three specific objectives:
(1) to systematically assess the assumptions underpinning the Emax model,
(2) to relax these assumptions to accommodate different experimental conditions and physiological behaviours of systems,
(3) to develop a user-friendly interface of the framework for scientific communication.
Methods: The assumptions underpinning the Emax model were identified based on an evaluation of its historical origins, subsequent mathematic derivations, expert opinion, and logical reasoning. PubMed, Scopus and Google Scholar were searched for the features of receptor pharmacology that were not in line with the assumptions of the Emax model. Subsequently, the publications that cited these papers were screened to identify necessary model components for describing these features. At the end, by assembling all the necessary model components together the Emax model was generalised into a cohesive model framework of receptor pharmacology. In addition, a Shiny website was developed for interactive presentation of the cohesive model framework.
Results: Seven assumptions underpinning the Emax model were identified:
Assumption 0: The ligand-receptor interaction follows the law of mass action.
Assumption 1: There is linear relationship between receptor occupancy and response.
Assumption 2: There is no ligand-independent receptor activity.
Assumption 3: One receptor only produces one type of response.
Assumption 4: The total amount of receptor is constant.
Assumption 5: The binding of ligand to receptor is at equilibrium.
Assumption 6: There is an excess of ligand.
Assumption 0: The law of mass action ensures that the reaction rate depends on the concentrations of the reactants or products and the stoichiometry, forming the foundation of drug action. This assumption seems to be valid for most cases.
Assumption 1: The Emax model cannot explain the phenomenon of receptor reserve (i.e., the ability of a ligand to elicit a maximal response with only a fraction of the whole receptor population occupied) [4], indicating a possible nonlinear relationship between receptor occupancy and response. Relaxation of Assumption 1 leads to the development of the operational model [5]. The operational model incorporates an arbitrary transduction function to transform receptor occupancy into response. Most of the time, it would be the rectangular hyperbolic function.
Assumption 2: The Emax model cannot explain the phenomenon of constitutive activity (i.e., ligand-independent receptor activity) or inverse agonist [6]. Relaxation of Assumption 2 leads to the two-state model [7]. In the two-state model, receptor could spontaneously form the active state and there exists dynamic equilibrium between active state and resting state even without any ligand present. In addition, the ligand could alter the equilibrium between resting state and active state. Note an empirical generalisation often includes a baseline effect as an approximation to the two-state model.
Assumption 3: The Emax model cannot explain the phenomenon of biased agonism (i.e., a ligand can act on one receptor to differentially regulate multiple signalling pathways) [8]. Relaxation of Assumption 3 leads to the three-state model (i.e., the simplest version of multi-state model) [9]. In three-state model, receptor has two mutually competing active states and therefore can have two distinct signalling profiles.
Assumptions 4-6: These Assumptions are related to the equilibrium conditions of the Emax model. The loss of surface receptor overtime has been observed in some receptors (e.g., cannabinoid 1 receptor and mu-Opioid receptor), suggesting the need to relax Assumption 4 and consider receptor turnover and internalisation [10,11]. The validity of Assumption 5 is largely depended on the relative magnitude between the drug-target residence time and the observation period. Because of the potential advantages on duration of pharmacological effect, there is an increasing interest in lead optimisation of long residence time [12]. The Emax model is not applicable for these ligands and ligand binding kinetic is warranted. As a pharmacometrician, we are more aware of the violation of Assumption 6. Due to ADME processes, the relative magnitude between the amount of receptor and the amount of ligand changes over time. Hence, a PK model of ligand is incorporated in most of modelling work. The relaxation of Assumptions 4-6 leads to target-mediated drug disposition model [13]. This model consists of receptor turnover, ligand binding kinetic, ligand-mediated receptor internalisation and ligand PK.
Generalising Assumptions 1-6 and integrating all the necessary model components provide a cohesive model framework of receptor pharmacology. Subsequently, a Shiny website was implemented for interactive presentation of this cohesive model framework (https://xiaozhu.shinyapps.io/GPCRmodel).
Conclusion: A single framework of receptor pharmacology is proposed as a series of generalisations of the standard Emax model which can accommodate different experimental conditions and physiological behaviours of systems. This framework allows modellers to examine their current use of the Emax model and facilitates the interpretation of modelling results. The next step of this work is to assess the identifiability of different sub-models from the cohesive model framework based on available input output data.
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