Mathematical modeling and application of genetic algorithm to parameter estimation in signal transduction: Trafficking and promiscuous coupling of G-protein coupled receptors

Abstract

G-protein-coupled receptors (GPCRs) constitute a large and diverse family of proteins whose primary function is to transduce extracellular stimuli into intracellular signals. These receptors play a critical role in signal transduction, and are among the most important pharmacological drug targets. Upon binding of extracellular ligands, these receptor molecules couple to one or several subtypes of G-protein which reside at the intracellular side of the plasma membrane to trigger intracellular signaling events. The question of how GPCRs select and activate a single or multiple G-protein subtype(s) has been the topic of intense investigations. Evidence is also accumulating; however, that certain GPCRs can be internalized via lipid rafts and caveolae. In many cases, the mechanisms responsible for this still remain to be elucidated. In this work, we extend the mathematical model proposed by Chen et al. [Modelling of signalling via G-protein coupled receptors: pathway-dependent agonist potency and efficacy, Bull. Math. Biol. 65 (5) (2003) 933–958] to take into account internalization, recycling, degradation and synthesis of the receptors. In constructing the model, we assume that the receptors can exist in multiple conformational states allowing for a multiple effecter pathways. As data on kinetic reaction rates in the signalling processes measured in reliable in vivo and in vitro experiments is currently limited to a small number of known values. In this paper, we also apply a genetic algorithm (GA) to estimate the parameter values in our model.

Introduction

Signal transduction is the process of conversion of external signals, such as hormones, growth factors, neurotransmitters and cytokines, to a specific internal cellular response, such as gene expression, cell division, or even cell suicide. This process begins at the cell membrane where an external stimulus initiates a cascade of enzymatic reactions inside the cell that typically includes phosphorylation of proteins as mediators of downstream processes [1]. Signal transduction consists of three processes. The first is reception—an agonist binds to a specific receptor on the cell membrane that triggers a change in the receptor molecule. The second is transduction. The change in the receptor brings about ordered sequences of biochemical reactions inside the cell that are carried out by enzymes and linked through second messengers. The third is response. After receiving the signal, target protein produces response which can be any of many different cellular activities, such as activation of a certain enzyme, rearrangement of the cytoskeleton, or changes in gene expression.

G-protein-coupled receptors (GPCRs) constitute a large and diverse family of proteins whose primary function is to transduce extracellular stimuli into intracellular signals. GPCRs are among the most heavily investigated drug targets in the pharmaceutical industry. They account for the majority of best-selling drugs and about 40% of all prescription pharmaceuticals on the market [2]. The inactive form of G-protein is a heterotrimer composed of three subunits, $\alpha ,\beta$ and $\gamma$ with a molecule of guanosine diphosphate (GDP) bound to the $\alpha$ subunit. The binding of ligands to receptors causes them to interact with the G-protein. The interaction of this inactive G-protein with bound receptor promotes the release of GDP from the $\alpha$ subunit and the binding of nucleotide guanosine triphosphate (GTP) at the same site. The G-protein is then released from the receptor and dissociates into separate $\beta \gamma$ and $\alpha$-GTP subunits. The $\alpha$-GTP is the active form of the G-protein. The activated $\beta \gamma$ and $\alpha$-GTP subunits in turn stimulate the generation of second messengers via intracellular effectors, passing on the signal by altering the activities of selected cellular proteins [3]. Depending on the type of G-protein to which the receptor is coupled, a variety of downstream signalling pathways can be activated [4], [5]. There are many examples of a single receptor coupling directly to more than one cellular signal transduction pathway [6], [7]. In traditional receptor theory, it is predicted that the relative degree of activation of each effector pathway by an agonist (relative efficacy) must be the same. More recently, however, evidence from a variety of systems suggests that some agonists, acting at a single receptor, may preferentially activate ${\mathrm{E}}_1$, say, while other agonists, acting at the same receptor in the same system, may preferentially activate ${\mathrm{E}}_2$ as shown schematically in Fig. 1 [6], [7]. These phenomenon termed “agonist-directed trafficking of receptor stimulus” were originally proposed in [8].

Many mathematical models have been proposed to describe agonist-directed trafficking of receptor stimulus and promiscuous coupling of receptors in the signal transduction process. In actual fact, the idea that a receptor can adopt more than one active state was derived from the concept of agonist-directed trafficking of a receptor stimulation to explain the ability of structurally diverse agonists to activate different G-protein-mediated signaling [6], [7], [8]. According to this model, each agonist is able to promote its own specific active receptor state, leading to an unlimited number of receptor conformations. Thus, different active states of the receptor may be associated with a particular G-protein. In contrast, Leff et al. suggested a three-state model where the receptor might exist in three states, an inactive $(R)$ and two active formations $(R^{*},R^{**})$, accounting for multiple G-protein coupling but limiting the number of active conformations [9]. They studied the agonist activity in two systems, one being an intact system in which receptors and G-protein are uniformly distributed and the other comprising of isolated pathways each with a distinct G-protein, operating independently from the other. Assuming that the efficacy of each pathway is proportional to the number of receptors activated, this model allows for pathway-dependent agonist efficacy and successfully simulates the differential activation of effector pathways observed previously with 5-HT${}_{2c}$ agonists [6]. Leff et al. [9] have; however, neglected the role of G-protein activation and, although different agonist potency has been predicted for the system of isolated pathways, their intact system has failed to predict the pathway-dependent agonist potency which has been observed in various experiments. Chen et al. [10] proposed a mathematical model which demonstrates the role of G-proteins in determining pathway-dependent agonist potency. In their model, the receptor can exist in four conformational states, one inactive and three active states. Chen et al. [10] have; however, neglected many biological factors concerning cell signalling via GPCRs, such as synthesis, degradation, internalization and recycling of receptors. Under normal physiological conditions, however, these dynamic trafficking events take place concurrently with receptor–ligand binding [11], [12], [13], [14].

While being a natural activity of receptors linked to signaling, internalization may be a therapeutically useful activity in itself. Ligands that selectively induce receptor internalization may have utility in the prevention of HIV-1 infection. This is because internalization may remove critical co-receptors for membrane fusion and subsequent HIV-1 infection [15], [16], [17]. In fact, this approach may be superior to blocking the HIV-1 infection. In many cases, the mechanisms responsible for these dynamic trafficking of receptors still remain to be elucidated. The focus of our present work is to study the effect of receptors trafficking, including receptors internalization, receptors synthesis, recycling of receptors and receptors degradation by extending a mathematical model proposed in [10].

As data on kinetic reaction rates in the signalling processes measured in reliable in vivo and in vitro experiments is currently limited to a small number of known values. In this paper, we also apply a genetic algorithm (GA) to estimate the parameter values in our model. The GA is an effective stochastic global search algorithm that is inspired by the evolutionary features of biological systems [18]. The GAs are the most popular evolutionary techniques, in virtue of their conceptual simplicity, the ease of programming entailed, and small number of parameters to be defined. Moreover, they have been shown to outperform alternative search techniques on difficult problems involving high dimensional, discontinuous, noisy and multi-modal objective functions [19], [20]. It has been successfully applied to various problems, such as function optimizations, parameter estimation in biochemical pathways [21], [22], [23], cancer gene search [24] and parameter estimation in mathematical modeling [20]. In the present work, a GA was applied to estimate 18 parameter values in our model and the predictions of the model were compared with the experimental results obtained by the authors of [7].

There are some quantitative pharmacological terms that are useful for our present analysis. Efficacy is defined as the ability of a drug to produce a stimulus, indicated by the maximum effect that can be produced by that drug. Potency, commonly expressed as the EC50, refers to the concentration or amount of an agonist needed to produce a 50% of the maximum effect of that agonist. A full agonist is a ligand that binds to a receptor and leads to a maximum biological response in the system under study while a partial agonist is an agonist that does not elicit as large an effect as a full agonist. An antagonist is a ligand that binds to a receptor, but does not produce a biological response, while blocking the actions of agonists. An inverse agonist means a ligand that binds to a receptor and reduces the constitutive activity of the receptor, thereby producing an effect opposite to that of an agonist.