Mathematical models for the EP2 and EP4 signaling pathways and their crosstalk

Abstract

G protein-coupled receptors EP2 and EP4 are both activated by the lipid messenger Prostaglandin E2 (PGE2) and induce the intracellular production of cyclic AMP (cAMP), ultimately affecting gene expression. Changes in cellular responses to PGE2 can have important consequences on immunity and disease, yet a detailed understanding of the EP2-EP4 signaling network is lacking. EP2 and EP4 are often co-expressed in cells but their specific contribution to cAMP production is poorly understood. Experimental data have shown that cAMP levels differ depending on whether PGE2 triggers EP2 or EP4, or both. To better understand the underlying mechanisms and predict cellular responses to PGE2, we developed mathematical models for EP2 and EP4 cAMP signaling, including receptor crosstalk. The mathematical models qualitatively reproduce the experimentally observed cAMP levels and provide mechanistic insight into both the differences and commonalities in EP2/EP4 signaling. We found that ligand binding dynamics play a crucial role for both single-receptor signaling and inter-receptor crosstalk. Inhibition of PGE2 signaling via receptor antagonists is gaining increasing attention in tumor immunology. These mathematical models could therefore contribute to the design of more effective anti-tumor therapies targeting EP2 and EP4.

1. Introduction

Prostaglandin E2 (PGE2) is a lipid mediator that plays an important role in modulating myeloid immune cells such as macrophages and dendritic cells (DCs) through the activation of the G protein-coupled receptors (GPCRs) EP2 and EP4 [1]. Both EP2 and EP4 signaling lead to an increase of intracellular cyclic adenosine monophosphate (cAMP) levels via ${\mathrm{G}}_{{\alpha }_s}$ protein activation. Additionally, EP4 signals through the inhibitory ${\mathrm{G}}_{{\alpha }_i}$ protein that decreases cAMP production. While the key players of these signaling pathways are known, the specific contributions of EP2 and EP4 and their potential crosstalk in directing cellular outcomes are poorly understood. We have previously shown experimentally that EP2 and EP4 signaling generate distinct cAMP production profiles [2]. While EP4 induced a transient cAMP response with a sharp initial increase and then decay over time, EP2 produced a sustained cAMP response only at high PGE2 levels. Furthermore, simultaneous EP2 and EP4 activation led to reduced cAMP production, compared to that by a single receptor yet retained a transient profile similar to EP4. These data collectively suggested the possibility of signaling crosstalk between EP2 and EP4, but the mechanisms of this crosstalk are not known. Mathematical models specifically dedicated to EP2-EP4 signaling would enable simulation of multiple signaling scenarios and improve predictions of cellular responses to PGE2.

There is an extensive literature on mathematical models, with varying level of detail and complexity, describing G protein activation cycles of general GPCRs, see e.g. [3, 4, 5], and cAMP regulation dependent on multiple subcellular enzymes, see e.g. [6, 7]. However, there is no unifying framework to describe the complete signaling pathways of GPCRs and many models assume that ligand-receptor binding is rapid. In particular, specific models for the cAMP signaling pathways of EP2 and EP4 that can predict and explain the experimentally observed cAMP production profiles in [2] are lacking. The cAMP signaling pathways consist of three steps: ligand binding to the receptor; receptor-mediated activation of heterotrimeric G proteins, and stimulation of adenylyl cyclase to synthesize cAMP. A mathematical model considering these three steps in the cAMP signaling cascade of the GLP-1R receptor was considered in [8]. The production of cAMP in response to EP2 activation was also modeled, with the assumption that ligand-receptor binding is rapid [9]. However, that model does not recapitulate the PGE2-induced EP2 threshold behavior of cAMP production that we experimentally observed [2].

Here, we develop mathematical models that mechanistically explain the commonalities and differences in EP2 and EP4 signaling and their crosstalk, and capture the experimentally observed PGE2-induced cAMP production profiles. The models are formulated as systems of ordinary differential equations and describe the dynamics of the ligand PGE2 activating the receptors, the intracellular G protein activation cycles and cAMP production dependent on the G proteins ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$. A key insight from the models is that receptor-specific ligand binding kinetics can explain the observed differences in the cAMP profiles of EP2 and EP4. The threshold behavior of EP2 in cAMP expression when varying PGE2 concentrations can be explained by higher order reactions in the binding between ligands and receptors. Inclusion of the temporally dynamic interactions between PGE2 and EP4, and the influence of receptor internalization, were essential to obtain the experimentally observed transient behavior of cAMP levels over time. Moreover, for EP4 it is important to take into account the competitive binding between the stimulating ${\mathrm{G}}_{{\alpha }_s}$ and the inhibiting ${\mathrm{G}}_{{\alpha }_i}$ proteins. The sensitivity of cAMP to the positive and negative feedback from ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$ proteins is also reflected in the level of cAMP when considering crosstalk between EP2 and EP4.

PGE2 is an important mediator in the immune system that can induce both activating and suppressive immune responses, for example against tumors [10]. As such, targeting PGE2 and its receptors is a promising therapeutic strategy to improve anti-tumor immune responses, but a better understanding of the signaling pathway is needed. The mathematical models developed in this study will contribute to unravel EP2 and EP4 specific signaling in order to help predict immune cell responses and the effect of drugs targeting these receptors.

The outline of our paper is as follows: in Section 2 we describe the three steps in the cAMP signaling cascade of EP2 and EP4 in greater detail and present the experimentally observed cAMP profiles of EP2, EP4 and their crosstalk in response to PGE2; in Section 3 we develop mathematical models for the three steps in the PGE-induced production of cAMP by EP2 and EP4; in Section 4 we present the numerical simulation results of the models, which are then discussed in Section 5.

2. Biological background and experimental results

2.1. Ligand binding

GPCR signaling is initiated through the binding of a ligand (e.g. neurotransmitter, chemokine or a lipid like PGE2) to the extracellular domain of the receptor that is typically embedded in the cell membrane. The receptor undergoes a conformational change and becomes activated. This allows the receptor to interact intracellularly with a nearby heterotrimeric G protein (composed of $\alpha ,\beta$, and $\gamma$ subunits) that will in turn exchange GDP for GTP on the ${\mathrm{G}}_{\alpha }$ subunit and will become active. The ${\mathrm{G}}_{{\alpha }^{GTP}}$ and ${\mathrm{G}}_{\beta \gamma }$ subunits will dissociate from the receptor and interact with different effector proteins (e.g. adenylyl cyclase, AC), triggering downstream signaling cascades such as cAMP production [5]. Depending on the specific ligand and GPCR the ligand-receptor binding kinetics can be significantly different. EP2 and EP4 both bind to the same ligand PGE2, but the ligand-receptor dynamics shows several qualitative differences. Here, we only mention the processes relevant for EP2 and EP4.

The activation/inactivation cycle of a receptor can display zero-order, ultrasensitivity behavior with respect to signals, i.e. a small change in the ligand concentration can lead to a large change in the concentration of activated G proteins [11]. The concentration of activated G proteins as a function of the ligand concentration then behaves switch-like, which resembles the responses of cooperative enzymes [12], and leads to a switch-like behavior in the cAMP profile as observed for EP2, but not for EP4. Furthermore, some GPCRs are internalized after ligand binding and can also be recycled back to the cell surface after signaling [13]. In the system studied here, it is known that EP4 is rapidly internalized upon ligand binding [2], while EP2 is not. Another difference between EP2 and EP4 concerns the affinity of the ligand for the receptors. Radioligand binding studies indicated that the affinity of PGE2 for EP4 receptors is higher than for EP2 [14] and the dissociation constant of PGE2-EP2 can be an order of magnitude larger than the dissociation constant of EP4 [15].

Denoting by $r$ the free receptors and by $r^{\mathrm{*}}$ the receptors bound to the ligand $p$, the reactions occurring upon ligand binding are as follows

$$\text{for}\text{EP2}\text{and}\text{EP4:}r+p⟷r^{\mathrm{*}},\text{for}\text{EP4:}{r}^{\mathrm{*}}⟶\text{internalized}\text{receptor.}$$ (1)

2.2. G protein activation cycle

We consider the (simplified) G protein activation cycle shown in Figure 1, see [4, 9, 5]. Once EP2 and EP4 are activated by PGE2 the heterotrimeric G protein complex ${\mathrm{G}}_{\alpha \beta \gamma }$ consisting of an $\alpha$-, $\beta$- and $\gamma$-subunit can bind to its cytosolic tail. The $\alpha$-subunit, in its inactive state, is also bound to guanosine diphosphate (GDP). When the complex ${\mathrm{G}}_{\alpha \beta \gamma }$ binds to the receptor the GDP-molecule dissociates and guanosine triphosphates (GTP) can bind to and activate the G protein. This causes the G protein complex to dissociate into ${\mathrm{G}}_{\beta \gamma }$ and the activated component ${\mathrm{G}}_{\alpha }$. The activated ${\mathrm{G}}_{\alpha }$ protein then hydrolyzes GTP to GDP and becomes inactivated. Finally, ${\mathrm{G}}_{\alpha }$ associates again with ${\mathrm{G}}_{\beta \gamma }$ to form the complex ${\mathrm{G}}_{\alpha \beta \gamma }$ and the cycle repeats.

Figure 1:

G protein cycle, modified from [4]. The activated receptor is denoted by ${\mathrm{R}}^{\mathrm{*}}$.

There are four major families of ${\mathrm{G}}_{\alpha }$ proteins, namely proteins with an ${\alpha }_s,{\alpha }_i,{\alpha }_q$ or ${\alpha }_{12/13}$-component, and different receptors activate different ${\mathrm{G}}_{\alpha }$ proteins. To model the G protein activation cycle we make the same simplifying assumptions as in [9], where the following reactions are labeled in Figure 1,

$$1:{\mathrm{G}}_{{\alpha }^{GDP}}+{\mathrm{G}}_{\beta \gamma }⟷{\mathrm{G}}_{{\alpha }^{GDP}\beta \gamma },$$ (2)

$$2:{\mathrm{G}}_{{\alpha }^{GDP}\beta \gamma }⟶{\mathrm{G}}_{{\alpha }^{GTP}}+{\mathrm{G}}_{\beta \gamma },$$ (3)

$$3:{\mathrm{G}}_{{\alpha }^{GTP}}⟶{\mathrm{G}}_{{\alpha }^{GDP}}.$$ (4)

Namely, we suppose that receptor activation and the association of receptors with $\mathrm{G}$ proteins are rapid compared to $\mathrm{G}$ protein activation, i.e. the second reaction takes place in a single step. Moreover, we suppose that ${\mathrm{G}}_{{\alpha }^{GTP}}$ has a negligible affinity for ${\mathrm{G}}_{\beta \gamma }$ compared to the affinity of ${\mathrm{G}}_{{\alpha }^{GDP}}$, such that ${\mathrm{G}}_{{\alpha }^{GTP}}$ does not bind to ${\mathrm{G}}_{\beta \gamma }$.

2.3. cAMP production

The activity of EP2 and EP4 can be measured experimentally by determining cAMP levels. In the EP2 and EP4 signaling pathways the cAMP concentration is mainly regulated by two enzymes, adenylyl cyclase (AC) and phosphodiesterase (PDE) [16]. Depending on the cell type and receptors present, different AC and PDE isoforms are expressed and specifically activated. AC and PDE isoforms show different regulation patterns [17], but little is known about which specific isoforms are activated in the cAMP signaling cascade of EP2 and EP4. Table 1 shows the stimulation and inhibition of certain G proteins for different AC isoforms.

Table 1:

Regulation patterns of AC-isoforms [17].

	G proteins
AC-isoforms	Stimulation	Inhibition
Group I
AC1	${\mathrm{G}}_{{\alpha }_s}$	${\mathrm{G}}_{\beta \gamma }$, ${\mathrm{G}}_{{\alpha }_i}$, ${\mathrm{G}}_{{\alpha }_z}$, ${\mathrm{G}}_{{\alpha }_o}$
AC8	${\mathrm{G}}_{{\alpha }_s}$	${\mathrm{G}}_{\beta \gamma }$
AC3	${\mathrm{G}}_{{\alpha }_s}$	${\mathrm{G}}_{\beta \gamma }$
Group II
AC2	${\mathrm{G}}_{{\alpha }_s}$, ${\mathrm{G}}_{\beta \gamma }$
AC4	${\mathrm{G}}_{{\alpha }_s}$, ${\mathrm{G}}_{\beta \gamma }$
AC7	${\mathrm{G}}_{{\alpha }_s}$, ${\mathrm{G}}_{\beta \gamma }$
Group III
AC5	${\mathrm{G}}_{{\alpha }_s}$, ${\mathrm{G}}_{\beta \gamma }$	${\mathrm{G}}_{{\alpha }_i}$, ${\mathrm{G}}_{{\alpha }_z}$
AC6	${\mathrm{G}}_{{\alpha }_s}$, ${\mathrm{G}}_{\beta \gamma }$	${\mathrm{G}}_{{\alpha }_i}$, ${\mathrm{G}}_{{\alpha }_z}$
Group IV
AC9	${\mathrm{G}}_{{\alpha }_s}$

AC enzymes are activated by ${\mathrm{G}}_{{\alpha }_s}$ and once activated, AC converts adenosine triphosphate (ATP) into cAMP [18]. On the other hand, ${\mathrm{G}}_{{\alpha }_i}$ inhibits AC [16] by preventing ${\mathrm{G}}_{{\alpha }_s}$-binding [19]. EP2 only activates ${\mathrm{G}}_{{\alpha }_s}$ while EP4 activates both, ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$ [2]. Hence, EP4 may also have an inhibitory effect on cAMP production, as shown in Figure 2. The other important regulators of cAMP in the EP2 and EP4 signaling pathways are PDE enzymes. They hydrolyze cAMP into adenosine monophosphate (AMP) and hence lead to a decrease in cAMP levels [16].

Figure 2:

Schematic representation of the PGE2 signaling receptors EP2 and EP4, reproduced and modified from [1]. EP2 and EP4 are GPCRs coexpressed in immune cells and bind PGE2. After binding to PGE2, both receptors activate the ${\mathrm{G}}_{{\alpha }_s}$ proteins, which in turn activate AC leading to increased levels of intracellular cAMP. EP4 can additionally activate ${\mathrm{G}}_{{\alpha }_i}$, leading to a modulation of cAMP production.

2.4. Experimental data

Figure 3 shows data on EP2 and EP4 signaling in response to PGE2 based on experiments with mouse RAW 264.7 macrophages expressing a cAMP FRET-biosensor [2]. The normalized FRET ratio is plotted against time (seconds). The FRET ratio measures the transmission of energy from a donor molecule to an acceptor molecule within the biosensor, such that when the biosensor binds to cAMP, the FRET ratio decreases. In Figure 3, the FRET ratio is seen to decrease with PGE2 addition, indicating an increase in cAMP concentration. The FRET ratio was measured for the PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M}$ and $10\mu \mathrm{M}$, as well as for a control experiment where only buffer was added. The FRET ratio is shown for the following cases:

1. EP2 and EP4 active

2. EP4 active, EP2 blocked by an antagonist

3. EP2 active, EP4 blocked by an antagonist

4. EP2 and EP4 blocked by antagonists

Figure 3:

Experimental data, reproduced from [2]. EP2 and EP4 induce distinct cAMP responses. The intramolecular cAMP FRET sensor t-Epac-vv was used: binding of cAMP to t-Epac-vv reduces Förster Resonance Energy Transfer (FRET) between the mTurquoise donor and Venus acceptor fluorophores of t-Epac-vv, making a decreased ratio of the fluorescent intensities a direct measure of cAMP accumulation. (A) FRET ratios of t-Epac-vv before and after the addition of different PGE2 concentrations were measured in transiently transfected RAW macrophages. A control was performed with the addition of buffer only. The data presented are mean ± SD from ≥ 5 cells per condition. FRET ratios were measured after the addition of PGE2 in cells pretreated with EP2 antagonist (ant.) AH6809 (B), EP4 antagonist GW627368X (C), or pretreated with both GW627368X and AH6809 (D). The data presented are mean ± SD from ≥4 cells per condition.

Note that only relative cAMP concentrations are shown; no quantitative measurements are available. The control experiment (black dots) is the baseline and the change in cAMP concentration is determined relative to that baseline. In Figure 3(D) we observe cAMP production for a PGE2 concentration of $10\mu \mathrm{M}$. A possible explanation is that the antagonist concentrations are too low for this high PGE level. This should be taken into account in Figures 3(A)–(C), i.e. for PGE2 concentrations of $10\mu \mathrm{M}$, EP2 and/or EP4 may not be fully blocked by the antagonists.

The data indicate significant qualitative differences in the signaling behavior of the receptors. Figure 3(C) displays that EP2 activity shows a clear threshold behavior with respect to PGE2. Namely, for the PGE2 concentrations $0.01\mu \mathrm{M}$ and $0.1\mu \mathrm{M}$, there is negligible cAMP production, while for the PGE2 concentrations $1\mu \mathrm{M}$ and $10\mu \mathrm{M}$, we observe a high production which stabilizes quickly. On the contrary, Figure 3(B) shows that the cAMP production for EP4 increases gradually with PGE2 levels. Moreover, we notice a clear peak in the cAMP concentration and then a decrease over time. The cAMP profiles for the combined signaling of EP2 and EP4 shown in Figure 3(A) are similar to EP4 but the peak cAMP levels are lower. Schematic plots are displayed in Figure 4.

Figure 4:

Schematic overview of the cAMP responses induced by EP2 and EP4, adapted from [2]. (A) When only EP4 is active, both ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$ control AC activity. ${\mathrm{G}}_{{\alpha }_s}$ induces a cAMP response that is dampened by ${\mathrm{G}}_{{\alpha }_i}$. The cAMP signal decays over time. (B) When EP2 is activated selectively, only ${\mathrm{G}}_{{\alpha }_s}$ modulates AC activity. The resulting cAMP response does not decay. (C) When both EP2 and EP4 are active, competition for ${\mathrm{G}}_{{\alpha }_s}$ dampens the integrated cAMP response. Signaling crosstalk between EP2 and EP4 allows the cell to respond differently to PGE2 depending on the organization and expression of EP2 and EP4. The drawings in this figure are a simplified representation of the experimental data and are mostly based on the more physiological PGE2 concentrations used (i.e. $0.01-1\mu M$)

3. Mathematical models

In this section we develop mathematical models for the cAMP signaling cascades of EP2 and EP4 and their crosstalk. We address ligand-receptor binding, G protein activation cycles and cAMP production separately.

3.1. Ligand binding

EP2 and EP4 both bind to the ligand PGE2, but the receptors have distinct structural characteristics and show different signaling modes. To take these differences into account and to develop a model for the crosstalk of the receptors we describe the dynamic interactions between ligands and receptors. To this end we use a simple model for ligand-receptor dynamics similar to [20] that allows us to include receptor-specific binding kinetics and replicates the characteristic EP2 and EP4 signaling responses, but limits the number of unknown parameters. Interactions between ligands, receptors and G proteins are often modeled using the cubic ternary complex model [21] or one of its generalizations, see e.g. [22]. These models consider additional states in receptor activation, e.g. receptor activation is modeled as a separate step and G proteins can also to bind to inactive receptors. We model the reactions in (1) and denote the concentrations of EP2 by $r_1$, of EP4 by $r_2$ and the concentration of PGE2 by $p$. We assume that EP2 binds to PGE2 at rate ${\mu }_1$ and EP4 at rate ${\mu }_2$. Once bound to PGE2 the receptors are in an activated state, denoted by $r_1^{\mathrm{*}}$ and $r_2^{\mathrm{*}}$, respectively. The dissociation rate for EP2 and PGE2 is ${\delta }_1$ and ${\delta }_2$ for EP4 and PGE2. EP4 shows internalization at rate ${\rho }_2$, while EP2 does not internalize, i.e. ${\rho }_1=0$. The dynamics of EP2 and EP4 activation through binding to the ligand PGE2 is then described by the system of ordinary differential equations

$$\begin{array}{l}\frac{dp}{dt}=-{\mu }_m{r}_m{p}^{n_m}+{\delta }_m{r}_m^{\mathrm{*}},\\ \frac{d{r}_m}{dt}=-{\mu }_m{r}_m{p}^{n_m}+{\delta }_m{r}_m^{\mathrm{*}},\\ \frac{d{r}_m^{\mathrm{*}}}{dt}={\mu }_m{r}_m{p}^{n_m}-{\delta }_m{r}_m^{\mathrm{*}}-{\rho }_m{r}_{m,}^{\mathrm{*}}\end{array}$$ (5)

where $m=1$ for EP2 and $m=2$ for EP4. The Hill coefficient for EP2 is $n_1\ge 3$ and $n_2=1$ for EP4.

The model reflects the qualitative differences between EP2 and EP4 in the ligand-receptor binding kinetics. While EP4 undergoes rapid internalization in response to PGE2, EP2 does not internalize [23, 2], i.e. ${\rho }_1=0,{\rho }_2>0$. Moreover, PGE2 has a higher affinity for EP4 than for EP2 which is reflected in the parameter values for the kinetic association and dissociation rates ${\mu }_m$ and ${\delta }_m,m=1,2$. Finally, EP2 shows ultrasensitivity with respect to PGE2 levels, i.e. a small change in ligand concentration can cause a large change in the concentration of activated G proteins [11]. This is modeled by a Hill coefficient $n_1\ge 3$ and leads to the threshold behavior in cAMP production in response to PGE2 shown in Figure 3. A possible explanation for ultrasensitivity is dimerization. It was shown that dimerization can result in positive cooperativity [24] and thus, in a Hill coefficient greater than one.

Different from related models for GPCR signaling pathways that assume ligand binding is rapid and use quasi-steady state assumptions [9, 3], we specifically model ligand dynamics. System (5) is similar to the model for ligand binding to normal and decoy receptors in [20], but adjusted to the characteristics of EP2 and EP4. More specifically, we take ultrasensitivity of EP2 into account by a Hill coefficient $n_1\ge 3$ and the fact that EP4 undergoes rapid internalization in response to PGE2 while EP2 does not internalize.

Since EP2 does not internalize, the total EP2 receptor concentration $r_1^{tot}=r_1+r_1^{\mathrm{*}}$ is preserved. Assuming that ligand binding dynamics is rapid leads to the approximation

$$r_1^{\mathrm{*}}=\frac{r_1^{tot}{p}^{n_1}}{K_D+p^{n_1}},\text{where}{K}_D=\frac{{\delta }_1}{{\mu }_1}.$$ (6)

This quasi-steady state assumption with $n_1=1$ was used to model the EP2 cAMP signaling pathway in [9]. However, a Hill coefficient $n_1=1$ reflects a gradual increase in cAMP levels and cannot predict the characteristic cAMP profiles in Figure 3 displaying a threshold behavior with respect to PGE2 concentrations. For EP2 the quasi-steady state assumption (6) with $n_1\ge 3$ leads to nearly the same cAMP profile for EP2 as the kinetic model (5) (see Figures 8 and 9), but this assumption is not suitable for EP4. In fact, taking ligand binding dynamics into account allows us to model internalization of EP4 which leads to a decay of cAMP expression over time, as well as to combine the models for the EP2 and EP4 signaling pathways and describe crosstalk of the receptors.

Figure 8:

Simulation of the EP2 signaling pathway; model (12) with parameter values in Tables 3–5 for PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

Figure 9:

Simulation of the EP2 signaling pathway with the quasi-steady state approximation (6) and parameter values in Tables 3–5 for PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

3.2. EP2 G protein activation cycle

EP2 only activates the $\mathrm{G}$ protein ${\mathrm{G}}_{{\alpha }_s}$. We consider the $\mathrm{G}$ protein activation cycle in Figure 1 and denote by $V_1$ the difference between reassociation and dissociation rate of ${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma }$ in (2), by $V_2$ the dissociation rate of ${\mathrm{G}}_{{\alpha }_s^{GTP}\beta \gamma }$ catalyzed by the activated receptor EP2^∗ in (3) and by $V_3$ the hydrolysis rate of GTP on ${\mathrm{G}}_{{\alpha }_s}$ in (4). Denoting the concentrations of ${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma },{\mathrm{G}}_{\beta \gamma },{\mathrm{G}}_{{\alpha }_s^{GTP}}$ and ${\mathrm{G}}_{{\alpha }_s^{GDP}}$ by ${\beta }_s,\beta ,{\alpha }_s^{\mathrm{*}}$ and ${\alpha }_s$, respectively, the G protein activation cycle for EP2 is modeled by the system of ordinary differential equations

$$\begin{array}{ll}\frac{d{\beta }_s}{dt}=V_1-V_2,& \frac{d{\alpha }_s^{\mathrm{*}}}{dt}=V_2-V_3,\\ \frac{d\beta }{dt}=V_2-V_1,& \frac{d{\alpha }_s}{dt}=V_3-V_1.\end{array}$$

Based on the Law of Mass Action for the rate $V_1$ we have

$$V_1=k_{\beta s}{\alpha }_s\beta -d_{\beta s}{\beta }_s,$$

where $k_{\beta s}$ denotes the ${\mathrm{G}}_{{\alpha }_s^{GDP}}-\beta \gamma$ association rate and $d_{\beta s}$ the ${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma }$ dissociation rate. The protein complex ${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma }$ binds to the activated receptor and then dissociates into ${\mathrm{G}}_{{\alpha }_s^{GTP}}$ and ${\mathrm{G}}_{\beta \gamma }$. We assume that the binding of GTP is rapid compared to the dissociation of GDP, so that the receptor catalyzes the transition from ${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma }$ to ${\mathrm{G}}_{{\alpha }_s^{GTP}}$ in a single step. This enzymatic reaction is modeled by the Michaelis-Menten rate equation

$$V_2=k_s{r}_1^{\mathrm{*}}\frac{{\beta }_s}{{\beta }_s+K_{r_{1}s}},$$

where $k_s$ is the ${\mathrm{G}}_{{\alpha }_s}$ activation rate, $r_1^{\mathrm{*}}$ the concentration of activated EP2 receptors determined by (5) and $K_{r_{1}s}$ the dissociation constant for EP2 and ${\mathrm{G}}_{{\alpha }_s^{GTP}\beta \gamma }$. Finally, the hydrolysis of GTP to GDP is described by

$$V_3=d_s{\alpha }_s^{\mathrm{*}},$$

where $d_s$ is the hydrolysis rate. Adding the respective equations we observe that the following relations for mass conservation hold,

$$\beta +{\beta }_s=b_1,{\alpha }_s+{\beta }_s+{\alpha }_s^{\mathrm{*}}=a_s,$$

where $b_1$ is the total concentration of ${\mathrm{G}}_{\beta \gamma }$-units and $a_s$ the total concentration of ${\mathrm{G}}_{{\alpha }_s}$-units. Using these relations leads to the reduced system

$$\begin{array}{l}\frac{d{\alpha }_s^{\mathrm{*}}}{dt}=k_s{r}_1^{\mathrm{*}}\frac{b_1-\beta }{b_1-\beta +K_{r_{1}s}}-d_s{\alpha }_s^{\mathrm{*}},\\ \frac{d\beta }{dt}=k_s{r}_1^{\mathrm{*}}\frac{b_1-\beta }{b_1-\beta +K_{r_{1}s}}+d_{\beta s}\left(b_1-\beta \right)-k_{\beta s}\beta \left(a_s-b_1+\beta -{\alpha }_s^{\mathrm{*}}\right).\end{array}$$ (7)

System (7) is based on the model for the G protein activation cycle of EP2 in [9]. However, here, we take ligand binding kinetics and ultrasensitivity into account, i.e. $r_1^{\mathrm{*}}$ denotes the concentration of activated EP2 receptors modeled by system (5). In [9] ligand binding was assumed to be rapid and the quasi-steady state approximation (6) with Hill coefficient $n_1=1$ was used. However, as discussed in the previous subsection taking $n_1=1$ does not resemble the threshold behavior of EP2 in response to PGE2, see Figure 3. Moreover, the ligand-receptor dynamics is essential for EP4 and for modeling crosstalk of the receptors.

Several models for the G protein activation cycle of general GPCRs have been developed, with varying level of detail and complexity. For an overview we refer to [3, 25, 5, 9]. Most models describe general mechanisms, but specific models for the G protein activation cycles for EP2 and EP4 are not available. An exception is [9] where the EP2 cAMP signaling pathway was modeled. Finally we remark that as in [9] we do not assume that the total concentration of ${\mathrm{G}}_{\beta \gamma }$-units $b_1$ and ${\mathrm{G}}_{{\alpha }_s}$-units $a_s$ coincide, which differs from most models [8, 4, 5]. Only a few studies have quantified the ratio between the total G protein concentrations $b_1$ and $a_s$, but evidence suggests that it is not equal to one [9].

3.3. EP4 G protein activation cycles

EP4 activates two different G proteins, ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$. We assume that EP4 responds to both G proteins in the same way and activation of ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$ occurs according to the G protein activation cycle shown in Figure 1. Furthermore, we assume that the protein complexes ${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma }$ and ${\mathrm{G}}_{{\alpha }_i^{GDP}\beta \gamma }$ are competing for EP4. We use the same notations as in Section 3.2 for the rates $V_1,V_2$ and $V_3$ describing the ${\mathrm{G}}_{{\alpha }_s}$ activation cycle, replacing EP2 by EP4. The association and dissociation rate of ${\mathrm{G}}_{{\alpha }_i^{GDP}\beta \gamma }$ is denoted by $V_4$, the dissociation rate of ${\mathrm{G}}_{{\alpha }_i^{GTP}\beta \gamma }$ catalyzed by the activated EP4 receptor by $V_5$ and the hydrolyses rate of GTP on ${\mathrm{G}}_{{\alpha }_i}$ by $V_6$. The ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$ activation cycles for EP4 are then described by the system of ordinary differential equations

$$\begin{array}{ll}\frac{d{\beta }_s}{dt}=V_1-V_2,& \frac{d{\beta }_i}{dt}=V_4-V_5,\\ \frac{d{\alpha }_s^{\mathrm{*}}}{dt}=V_2-V_3,& \frac{d{\alpha }_i^{\mathrm{*}}}{dt}=V_5-V_6,\\ \frac{d{\alpha }_s}{dt}=V_3-V_1,& \frac{d{\alpha }_i}{dt}=V_6-V_4,\\ \frac{d\beta }{dt}=V_2-V_1+V_5-V_4,& \end{array}$$ (8)

where ${\beta }_i,{\alpha }_i^{\mathrm{*}}$ and ${\alpha }_i$ are the concentrations of ${\mathrm{G}}_{{\alpha }_i^{GDP}\beta \gamma },{\mathrm{G}}_{{\alpha }_i^{GTP}}$ and ${\mathrm{G}}_{{\alpha }_i^{GDP}}$, respectively. The rates $V_1$ and $V_3$ for the ${\mathrm{G}}_{{\alpha }_s}$-activation cycle are the same as for EP2, and correspondingly, for ${\mathrm{G}}_{{\alpha }_i}$ we have

$$V_4=k_{\beta i}{\alpha }_i\beta -d_{\beta i}{\beta }_i,V_6=d_i{\alpha }_i^{\mathrm{*}},$$

where $k_{\beta i}$ denotes the ${\mathrm{G}}_{{\alpha }_i}^{GDP}-\beta \gamma$ association rate, $d_{\beta i}$ the ${\mathrm{G}}_{{\alpha }_i\beta \gamma }$ dissociation rate and $d_i$ the hydrolysis rate. In the G protein activation by EP4 we take competitive binding into account, i.e. we assume that if the complex ${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma }$ is bound to the receptor it cannot bind to ${\mathrm{G}}_{{\alpha }_i^{GDP}\beta \gamma }$, and vice versa. This is modeled by the rates

$$V_2=k_s{r}_2^{\mathrm{*}}\frac{{\beta }_s}{{\beta }_s+K_{r_{2}s}+\frac{K_{r_{2}s}}{K_{r_{2^i}}}{\beta }_i},V_5=k_i{r}_2^{\mathrm{*}}\frac{{\beta }_i}{{\beta }_i+K_{r_{2}i}+\frac{K_{r_{2}s}}{K_{r_{2^i}}}{\beta }_s},$$

where $k_s$ and $k_i$ are the ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$ activation rates, $r_2^{\mathrm{*}}$ the concentration of activated EP4 receptors determined by (5) and $K_{r_{2}s}$ and $K_{r_{2}i}$ the EP4 dissociation constants for ${\mathrm{G}}_{{\alpha }_s^{GTP}\beta \gamma }$ and ${\mathrm{G}}_{{\alpha }_i^{GTP}\beta \gamma }$, respectively. These rates are an approximation for competitive binding which is derived as follows. For the concentration of activated EP4 receptors $r_2^{\mathrm{*}}$ we have

$$r_2^{\mathrm{*}}=r_{2s}^{\mathrm{*}}+r_{2i}^{\mathrm{*}}+r_{20}^{\mathrm{*}},$$ (9)

where $r_{2s}^{\mathrm{*}}$ and $r_{2i}^{\mathrm{*}}$ are the concentrations of activated EP4 bound to ${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma }$ and ${\mathrm{G}}_{{\alpha }_i^{GDP}\beta \gamma }$, respectively, and $r_{20}^{\mathrm{*}}$ is the concentration of activated EP4 unbound to G protein complexes. We model the binding of the G protein complexes to the receptor by the Michaelis-Menten rate equations

$$r_{2s}^{\mathrm{*}}=\frac{r_{20}^{\mathrm{*}}{\beta }_s}{K_{r_{2}s}+r_{20}^{\mathrm{*}}},r_{2i}^{\mathrm{*}}=\frac{r_{20}^{\mathrm{*}}{\beta }_i}{K_{r_{2}i}+r_{20}^{\mathrm{*}}}.$$

Assuming that $K_{r_{2}s}\gg r_{20}^{\mathrm{*}}$ and $K_{r_{2}i}\gg r_{20}^{\mathrm{*}}$ we can approximate these rates by

$$r_{2s}^{\mathrm{*}}\approx \frac{r_{20}^{\mathrm{*}}{\beta }_s}{K_{r_{2}s}},r_{2i}^{\mathrm{*}}\approx \frac{r_{20}^{\mathrm{*}}{\beta }_i}{K_{r_{2}i}}.$$

Using these relations in (9) we obtain

$$r_{20}^{\mathrm{*}}\approx \frac{r_2^{\mathrm{*}}}{1+\frac{{\beta }_s}{K_{r_{2}s}}+\frac{{\beta }_i}{K_{r_{2}i}}},$$

which leads to the approximate rates used in $V_2$ and $V_5$,

$$r_{2s}^{\mathrm{*}}\approx \frac{r_2^{\mathrm{*}}{\beta }_s}{{\beta }_s+K_{r_{2}s}+\frac{K_{r_{2}s}}{K_{r_{2^i}}}{\beta }_i},r_{2i}^{\mathrm{*}}\approx \frac{r_2^{\mathrm{*}}{\beta }_i}{{\beta }_i+K_{r_{2}i}+\frac{K_{r_{2}s}}{K_{r_{2}i}}{\beta }_s}.$$

Adding the corresponding equations in System (8) we observe that the following mass conservation assumptions hold

$$\beta +{\beta }_s+{\beta }_i=b_2,{\alpha }_s+{\beta }_s+{\alpha }_s^{\mathrm{*}}=a_s,{\alpha }_i+{\beta }_i+{\alpha }_i^{\mathrm{*}}=a_i,$$

where $b_2$ is the total concentration of ${\mathrm{G}}_{\beta \gamma }$-units and $a_s$ and $a_i$ are the total concentrations of ${\mathrm{G}}_{{\alpha }_s}$-units and ${\mathrm{G}}_{{\alpha }_i}$-units respectively. Using these relations we can reduce the model to the system

$$\begin{array}{l}\frac{d{\alpha }_s^{\mathrm{*}}}{dt}=k_s{r}_2^{\mathrm{*}}\frac{b_2-\beta -{\beta }_i}{b_2-\beta -{\beta }_i+K_{r_{2}s}+\frac{K_{r_{2}s}}{K_{r_{2^i}}}{\beta }_i}-d_s{\alpha }_s^{\mathrm{*}},\\ \frac{d{\alpha }_i^{\mathrm{*}}}{dt}=k_i{r}_2^{\mathrm{*}}\frac{{\beta }_i}{{\beta }_i+K_{r_{2}i}+\left(b_2-\beta -{\beta }_i\right)\frac{K_{r_{2}i}}{K_{r_{2}s}}}-d_i{\alpha }_i^{\mathrm{*}},\\ \frac{d{\beta }_i}{dt}=-k_i{r}_2^{\mathrm{*}}\frac{{\beta }_i}{{\beta }_i+K_{r_{2}i}+\left(b_2-\beta -{\beta }_i\right)\frac{K_{r_{2}i}}{K_{r_{2}s}}}+k_{\beta i}\beta \left(a_i-{\beta }_i-{\alpha }_i^{\mathrm{*}}\right)-d_{\beta i}{\beta }_i,\\ \frac{d\beta }{dt}=k_i{r}_2^{\mathrm{*}}\frac{{\beta }_i}{{\beta }_i+K_{r_{2}i}+\left(b_2-\beta -{\beta }_i\right)\frac{K_{r_{2}i}}{K_{r_{2}s}}}+k_s{r}_2^{\mathrm{*}}\frac{b_2-\beta -{\beta }_i}{b_2-\beta -{\beta }_i+K_{r_{2}s}+\frac{K_{r_{2}s}}{K_{r_{2}i}}{\beta }_i}\\ +d_{\beta i}{\beta }_i+d_{\beta s}\left(b_2-\beta -{\beta }_i\right)-\beta \left(k_{\beta s}\left(a_s-b_2+\beta +{\beta }_i-{\alpha }_s^{\mathrm{*}}\right)+k_{\beta i}\left(a_i-{\beta }_i-{\alpha }_i^{\mathrm{*}}\right)\right).\end{array}$$ (10)

Most models for GPCRs describe receptor activation by one single ligand triggering one single pathway, i.e. the binding and activation of one specific G protein [4, 5, 8, 9]. A model for the binding of two different ligands to a receptor, such as agonist–antagonist competition, has been considered in [26]. Very few models take the activation of multiple downstream pathways into account. An equilibrium model describing receptor activation and the binding of the receptor to two different G proteins was proposed in [27]. More recently, a time-dependent multiple cubic ternary complex model for a receptor that can activate multiple G protein-mediated pathways was considered in [28]. We use the simpler system (10) to describe EP4, a GPCR activating two G proteins. It is an extension of the single G protein activation model (7) for EP2 and describes the EP4 activation cycles for ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$ taking the effects of competitive binding into account.

3.4. cAMP production

The production of cAMP in the EP2 and EP4 signaling pathways is mainly regulated by two enzymes, AC and PDE. Different isoforms of AC and PDE exist and depending on the cell type and organism multiple isoforms might be present in a cell. For the different regulation patterns of AC-isoforms we refer to Table 1. AC activation and cAMP degradation by PDE in response to EP2 and EP4 is not yet very well understood. For instance, it is known that both receptors activate AC2, but not AC3 [29]. In cells similar to mouse RAW 264.7 macrophages the isoforms AC2, AC3, AC6, AC7, and AC9 were found [30, 31]. Concerning the degradation by PDEs, the isoform PDE4 is considered the principal enzyme constraining cAMP signaling [29, 9].

Since information about the specific AC and PDE isoforms is not available and to reduce the number of unknown parameters in the model, we assume that one dominant AC-isoform is responsible for cAMP production, for which ${\mathrm{G}}_{{\alpha }_s}$ has a stimulating and ${\mathrm{G}}_{{\alpha }_i}$ an inhibiting effect, and that the isoform PDE4 dominates cAMP degradation.

For EP2 that only activates ${\mathrm{G}}_{{\alpha }_s}$ we use a Michaelis-Menten reaction function to model AC activation by activated ${\mathrm{G}}_{{\alpha }_s}$,

$$\frac{{\alpha }_s^{\mathrm{*}}{A}^{tot}}{{\alpha }_s^{\mathrm{*}}+K_{as}},$$

where $A^{tot}$ denotes the total AC concentration and $K_{as}$ the dissociation constant for AC and ${\mathrm{G}}_{{\alpha }_s}$. EP4 does not only activate ${\mathrm{G}}_{{\alpha }_s}$, but also ${\mathrm{G}}_{{\alpha }_i}$ which has an inhibiting effect on cAMP production. More specifically, ${\mathrm{G}}_{{\alpha }_i}$ proteins can bind to AC preventing the binding of ${\mathrm{G}}_{{\alpha }_s}$ to AC and hence, inhibit cAMP production. To take inhibition by ${\mathrm{G}}_{{\alpha }_i}$ into account we use the same approximation as in Section 3.3 for the competitive binding of ${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma }$ and ${\mathrm{G}}_{{\alpha }_i^{GDP}\beta \gamma }$ to EP4. Denoting by $AC$ the concentration of AC unbound to activated G proteins we have

$$A{C}^{\mathrm{t}\mathrm{o}\mathrm{t}}=\frac{AC{\alpha }_s^{\mathrm{*}}}{K_{as}+AC}+AC+\frac{AC{\alpha }_i^{\mathrm{*}}}{K_{ai}+AC},$$

where $K_{ai}$ is the dissociation constant for AC and ${\mathrm{G}}_{{\alpha }_i}$. Assuming that $K_{as}\gg AC$ and $K_{ai}\gg AC$ leads to the approximation

$$A{C}^{\mathrm{t}\mathrm{o}\mathrm{t}}\approx \frac{AC{\alpha }_s^{\mathrm{*}}}{K_{as}}+AC+\frac{AC{\alpha }_i^{\mathrm{*}}}{K_{ai}},$$

and we obtain

$$AC=\frac{A{C}^{\mathrm{t}\mathrm{o}\mathrm{t}}}{1+{\alpha }_s^{\mathrm{*}}/K_{as}+{\alpha }_i^{\mathrm{*}}/K_{ai}}.$$

Hence, the approximate cAMP production rate is given by

$$\frac{{\alpha }_s^{\mathrm{*}}{A}^{tot}}{{\alpha }_s^{\mathrm{*}}+K_{as}\left(1+{\alpha }_i^{\mathrm{*}}/K_{ai}\right)},$$

which takes inhibition by ${\mathrm{G}}_{{\alpha }_i}$ into account. PDE enzymes catalyze the decomposition of cAMP into AMP which we model by

$$\frac{w{P}^{tot}}{w+K_w},$$

where $w$ denotes the cAMP concentration, $P^{tot}$ is the total PDE4 concentration and $K_w$ is the dissociation constant for PDE4 and cAMP. Since relative concentrations are measured in the experiments, see Figure 3, we assume that there is no basal cAMP production rate. Hence, cAMP production is described by the ordinary differential equation

$$\frac{dw}{dt}=k_w\frac{{\alpha }_s^{\mathrm{*}}{A}^{tot}}{{\alpha }_s^{\mathrm{*}}+K_{as}+\eta K_{as}\frac{{\alpha }_i^{\mathrm{*}}}{K_{ai}}}-d_w{P}^{tot}\frac{w}{w+K_w},$$ (11)

where $k_w$ is the active cAMP production rate and $d_w$ the PDE4 catalyzed cAMP degradation rate. Moreover, through the parameter $\eta$ we can take inhibition by ${\mathrm{G}}_{{\alpha }_i}$ into account, i.e. $\eta =0$ for EP2 and $\eta =1$ for EP4.

Several models exist for cAMP regulation in GPCR signaling pathways, and many take into account additional processes of the signaling pathway [6, 7, 9, 32]. Here, we only consider the main enzymes, AC and PDE, that are responsible for cAMP production and degradation in EP2 and EP4 signaling. We make the simplifying assumption that for each of these enzymes there is one dominant isoform. The experimental data informing our model is qualitative and information is not available about the specific AC/PDE isoforms responsible in our cell type, which would be required for a more refined modeling approach. Another difference between our model and others [6, 7, 9] is that the experimental data reported cAMP profiles relative to a control experiment where no ligand is added so no basal production rate is included in (11). We took the inhibiting effect of ${\mathrm{G}}_{{\alpha }_i}$ into account based on competitive binding which is modeled in the same way as the competitive binding of ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$ to EP4. Instead of an approximation as we use here an explicit expression for the effects of competitive binding was considered in [33]. Another more detailed model for inhibition by ${\mathrm{G}}_{{\alpha }_i}$ including the possibility of dimerization and of the simultaneous binding of two ${\mathrm{G}}_{\alpha }$ molecules to AC5 was proposed in [34]. Even with these simplified assumptions, our model was able to correctly capture the qualitative cAMP dynamics.

For convenience of the reader we list in Table 2 the model variables with a description.

Table 2:

Description of model variables

	model	symbol	description
ligand receptor binding	(5)	$p$	PGE2 concentration
		$r_{1/2}$	inactive EP2/EP4 concentration
		$r_{1/2}^{\mathrm{*}}$	activated EP2/EP4 concentration
$G$ protein activation cycles	(7),(10)	${\alpha }_{s/i}$	$G_{{\alpha }_{s/i}^{GDP}}$ concentration
		${\alpha }_{s/i}^{\mathrm{*}}$	$G_{{\alpha }_{s/i}^{GTP}}$ concentration
		$\beta$	$G_{\beta \gamma }$ concentration
		${\beta }_{s/i}$	$G_{{\alpha }_{s/i}^{GDP}\beta \gamma }$ concentration
cAMP production	(11)	$w$	cAMP concentration

3.5. Parameter values

For only some of the parameters in equations (5), (7), (10) and (11) precise values are available for the receptors EP2 and EP4 and the specific cell types used in the experiments. The values used in the simulations in Section 4 are listed in Tables 3–5. If a parameter range is given the value used is indicated in brackets.

Table 3:

Parameter values for ligand dynamics and receptor activation

symb.	parameter	value	reference
${\mu }_1$	assoc. rate EP2 - PGE2	0.2417(sμM)−1	[15] (calculated from $K_d$, mouse)
${\mu }_2$	assoc. rate EP4 - PGE2	1.385(sμM)−1	[36] (human embryo. kidney cells)
${\delta }_1$	dissoc. rate EP2 - PGE2	0.0029 s−1	[20] (CD95L, DcR3 decoy receptor)
${\delta }_2$	dissoc. rate EP4 - PGE2	0.03325 s−1	[36] (human embryo. kidney cells)
${\rho }_1$	internalization rate EP2	0	assumed
${\rho }_2$	internalization rate EP4	0.00538 s−1	[23] (estimat., human 293-EBNA)
$n_1/n_2$	Hill coefficient EP2 /EP4	3/1	assumed
$r_1^{tot}$	total EP2 concentration	2–5nM (4nM)	[37, 4] (human T-cells)
$r_2^{tot}$	total EP4 concentration	8nM	assumed
$\sigma$	binding ratio for crosstalk	0.5	assumed

Table 5:

Parameter values for cAMP production

symb.	parameter	value	reference
$k_w$	active cAMP production rate	6.713s−1	[9] (modif. for AC6)
$K_{as}$	dissociat. const for AC6 and $G_{{\alpha }_s}$	0.2μM	[34]
$K_{ai}$	dissociat. const for AC6 and $G_{{\alpha }_i}$	0.027μM	[34]
$d_w$	cAMP degradation rate by PDE4	8.66s−1	[32]
$K_w$	dissociat. const for PDE4 and cAMP	1.21μM	[32]
$A^{tot}$	total AC concentration	0.0497μM	[6]
$P^{tot}$	total PDE concentration	0.039μM	[6]

3.5.1. Ligand-receptor binding

To model the threshold behavior of EP2 with respect to PGE2 we assume that EP2 shows ultrasensitivity and take the Hill coefficient $n_1=3$. For EP4 cAMP levels increase gradually with respect to PGE2 concentrations and hence, for EP4 we take $n_2=1$. While EP2 does not internalize, i.e. ${\rho }_1=0$, internalization plays a crucial role for EP4 [23, 35] and leads to the decrease of cAMP levels over time in Figure 3. The internalization rate ${\rho }_2$ is calculated based on [23] where internalization of EP4 was measured when exposed to $1\mu \mathrm{M}$ PGE2. It was found that EP4 receptors underwent rapid internalization to the extent of 40% with a half-time of approximately 2 minutes.

EP4 typically shows a higher binding affinity to PGE2 than EP2. Based on [15] the equilibrium dissociation constants $K_D$ are 12nM for PGE2 and EP2 and 1.9nM for PGE2 and EP4 (mouse). Taking ligand binding dynamics into account requires the kinetic association and dissociation rates for ligand and receptor, but most studies focus on the equilibrium dissociation constants $K_D$ or inhibitor constants $K_i$ [15]. The kinetic association and dissociation rates are not available in the literature for EP2 and EP4 and the cell type used in the experiments. We therefore use the association and dissociation rates for PGE2 and EP4 in [36] for human embryonic kidney cells. For PGE2 and EP2 kinetic association and dissociation rates are not available and therefore, we use the dissociation rate ${\delta }_1$ for DcR3 receptors and the ligand CD95L in [20] and calculate the association rate ${\mu }_1$ based on the relation $K_D={\delta }_1/{\mu }_1$ and the value $K_D=12\mathrm{n}\mathrm{M}$ for PGE2 and EP2 in [15]. We remark that for EP4 the relation $K_D={\delta }_2/{\mu }_2$ and parameter values in [36] lead to $K_D=0.02401\mu \mathrm{M}$, which is an order of magnitude higher than the equilibrium dissociation constant $K_D=1.9\mathrm{n}\mathrm{M}$ in [15]. Effects due to internalization could explain this difference, since the rate $K_D$ reflects an equilibrium dissociation rate.

The total receptor concentration $r_1^{tot}$ for EP2 is taken from [4, 37]. Moreover, we assume that the total concentration of EP4 is equal or twice as high as of EP2, as indicated in [2]. To model crosstalk, we combine the models for the ligand receptor binding for EP2 and EP4. Both receptors bind to the same ligand PGE2. We assume that the affinity of PGE2 for EP2 is the same as for EP4. Hence, in model (14) for the combined activation of EP2 and EP4 we take for the parameter controlling the affinity $\sigma =0.5$ reflecting that PGE2 has equal affinity for both receptors.

As initial values in the simulations we take the PGE2 concentrations used in the experiments and assume that initially, no receptor is activated, i.e.,

$$p\left(0\right)\in \left\{0.01\mu M,0.1\mu M,1\mu M,10\mu M\right\},r_m^{\mathrm{*}}\left(0\right)=0,r_m\left(0\right)=r_m^{tot},\text{for}m=1,2.$$

3.5.2. G protein activation cycles

The model for the G protein activation cycle of EP2 (7) and the extended model for EP4 (10) are based on the model for the signaling pathways of EP2 and C5a in [9]. We take the parameter values in this reference for EP2 and the reaction rates in the G protein activation cycles. The dissociation constants for EP4 and the G protein complexes ${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma }$ and ${\mathrm{G}}_{{\alpha }_i^{GDP}\beta \gamma }$ are taken from [38]. As in [9] we assume that the association rates for $G_{{\alpha }_s}$ and $G_{\beta \gamma }$ and for $G_{{\alpha }_i}$ and $G_{\beta \gamma }$ are the same. For EP4 inhibition by ${\mathrm{G}}_{{\alpha }_i}$ is taken into account in (10) through competitive binding and determined by the ratio of the dissociation constants $K_{r_{2}s}/K_{r_2}$.

We remark that the total ${\mathrm{G}}_{\beta \gamma }$ concentration $\beta {\gamma }^{tot}$ in [9] is based on the mass conservation assumption

$$\beta +{\beta }_s+{\beta }_i=\beta {\gamma }^{tot}.$$

As EP2 does not activate the G protein ${\mathrm{G}}_{{\alpha }_i}$ we neglect ${\mathrm{G}}_{{\alpha }_i^{GDP}\beta \gamma }$ in model (7) and therefore use a lower value for $\beta {\gamma }^{tot}$ for EP2. Here, we use $b_1=\beta {\gamma }^{tot}=5\mathrm{n}\mathrm{M}$ as it matches the initial value for ${\mathrm{G}}_{{\alpha }_i^{GDP}\beta \gamma }$ taken for EP4 while we take $b_2=\beta {\gamma }^{tot}=1.8\mu \mathrm{M}$ for EP4.

To model crosstalk of the receptors we combine the models (7) and (10). Both receptors activate the G protein ${\mathrm{G}}_{{\alpha }_s}$. We assume that the affinity of ${\mathrm{G}}_{{\alpha }_s}$ for the receptors EP2 and EP4 is the same, leading to the binding ratio ${\sigma }_0=0.5$.

As initial values we take the basal steady-state concentrations in absence of PGE2 obtained by running the corresponding model in the absence of PGE2,

$${\alpha }_s^{\mathrm{*}}\left(0\right)=0,\beta \left(0\right)=0.06nM,$$

and for EP4 we have, in addition,

$${\alpha }_i^{\mathrm{*}}\left(0\right)=0,{\beta }_i\left(0\right)=1.795\mu M.$$

3.5.3. cAMP production

The experimental data show inhibition by ${\mathrm{G}}_{{\alpha }_i}$ in the cAMP signaling cascade of EP4 [2]. Among the AC-isoforms found in cells similar to mouse RAW 264.7 macrophages, AC6 is the only one that is inhibited by ${\mathrm{G}}_{{\alpha }_i}$, see Table 1. Hence, we assume that AC6 is the dominant isoform for cAMP production. The parameter values for AC6 are taken from [34]. Here, we assume that AC6 has the same molecular weight as AC2 given in [9], namely 106 kDa. We also assume as in [9], that AC6 constitutes 0.1% of transfected S49 membranes which allows to rewrite the active cAMP production from 3.8 nmol/min mg into 6.713 s⁻¹. Concerning cAMP degradation by PDEs we assume that PDE4 is the primary isoform constraining cAMP expression. The parameter values for PDE4 are taken from [32].

We assume that initially, no receptor is activated and hence no cAMP is in the system, since cAMP levels are measured relative to a control experiment, i.e. as initial value we take

$$w\left(0\right)=0.$$

4. Simulations and results

In Subsection 4.1 we present the full models for the cAMP signaling pathways for EP2, EP4 and their crosstalk. Subsequently, we show in numerical simulations that the models qualitatively predict the experimentally observed cAMP levels in response to PGE2 shown in Figures 3 and 4. Numerical codes for simulations of model equations are implemented in Python 3.10.12, taking advantage of the Scipy 1.8.0 module [39, 40] and solutions for systems of ordinary differential equations were obtained using the scipy.integrate.odeint package with default parameters, i.e. the Jacobian functions are estimated numerically, relative tolerance = absolute tolerance = 1.49012 · 10⁻⁸, allowed maximum order for the non-stiff (Adams) method is 12 and for the stiff (BDF) method is 5, the initial time step size is estimated by the solver, with no lower or upper limits on the step size, and the sequence of time points for which the ordinary differential equations are solved is $t_k=k/15$, for $k=0,1,\dots ,3000$. The simulation results were plotted using Matplotlib 3.5.1 Python library [41].

4.1. Full models for signaling pathways

The full PGE2 induced cAMP signaling cascade for EP2 is described by the following system of ordinary differential equations combining the model for ligand-receptor binding (5) (with $m=1$) with the models for the EP2 G protein activation cycle (7) and cAMP production (11) (with $\eta =0$),

$$\begin{array}{l}\frac{dp}{dt}=-{\mu }_1{r}_1{p}^{n_1}+{\delta }_1{r}_1^{\mathrm{*}},\frac{d{r}_1}{dt}=-{\mu }_1{r}_1{p}^{n_1}+{\delta }_1{r}_1^{\mathrm{*}},\frac{d{r}_1^{\mathrm{*}}}{dt}={\mu }_1{r}_1{p}^{n_1}-{\delta }_1{r}_1^{\mathrm{*}},\\ \frac{d{\alpha }_s^{\mathrm{*}}}{dt}=k_s{r}_1^{\mathrm{*}}\frac{b_1-\beta }{b_1-\beta +K_{r_{1}s}}-d_s{\alpha }_s^{\mathrm{*}},\\ \frac{d\beta }{dt}=k_s{r}_1^{\mathrm{*}}\frac{b_1-\beta }{b_1-\beta +K_{r_{1}s}}+d_{\beta s}\left(b_1-\beta \right)-k_{\beta s}\beta \left(a_s-b_1+\beta -{\alpha }_s^{\mathrm{*}}\right),\\ \frac{dw}{dt}=k_w{A}^{tot}\frac{{\alpha }_s^{\mathrm{*}}}{{\alpha }_s^{\mathrm{*}}+K_{as}}-d_w\frac{w{P}_2^{tot}}{w+K_w}.\end{array}$$ (12)

The full PGE2 induced cAMP signaling cascade for EP4 is described by the following system of ordinary differential equations combining the model for ligand-receptor binding (5) (with $m=2$) with the models for the EP4 G protein activation cycles (10) and cAMP production (11) (with $\eta =1$),

$$\begin{array}{l}\frac{dp}{dt}=-{\mu }_2{r}_{2}p+{\delta }_2{r}_2^{\mathrm{*}},\frac{d{r}_2}{dt}=-{\mu }_2{r}_{2}p+{\delta }_2{r}_2^{\mathrm{*}},\frac{d{r}_2^{\mathrm{*}}}{dt}={\mu }_2{r}_{2}p-{\delta }_2{r}_2^{\mathrm{*}}-{\rho }_2{r}_2^{\mathrm{*}},\\ \frac{d{\alpha }_s^{\mathrm{*}}}{dt}=k_s{r}_2^{\mathrm{*}}f\left({\beta }_i,\beta \right)-d_s{\alpha }_s^{\mathrm{*}},\frac{d{\alpha }_i^{\mathrm{*}}}{dt}=k_i{r}_2^{\mathrm{*}}g\left({\beta }_i,\beta \right)-d_i{\alpha }_i^{\mathrm{*}},\\ \frac{d{\beta }_i}{dt}=-k_i{r}_2^{\mathrm{*}}g\left({\beta }_i,\beta \right)+k_{\beta i}\beta \left(a_i-{\beta }_i-{\alpha }_i^{\mathrm{*}}\right)-d_{\beta i}{\beta }_i,\\ \frac{d\beta }{dt}=k_i{r}_2^{\mathrm{*}}g\left({\beta }_i,\beta \right)+k_s{r}_2^{\mathrm{*}}f\left({\beta }_i,\beta \right)+d_{\beta i}{\beta }_i+d_{\beta s}\left(b_2-\beta -{\beta }_i\right)\\ -\beta \left(k_{\beta s}\left(a_s-b_2+\beta +{\beta }_i-{\alpha }_s^{\mathrm{*}}\right)+k_{\beta i}\left(a_i-{\beta }_i-{\alpha }_i^{\mathrm{*}}\right)\right),\\ \frac{dw}{dt}=k_w\frac{{\alpha }_s^{\mathrm{*}}{A}^{tot}}{{\alpha }_s^{\mathrm{*}}+K_{as}\left(1+{\alpha }_i^{\mathrm{*}}/K_{ai}\right)}-d_w\frac{w{P}^{tot}}{w+K_w},\end{array}$$ (13)

where

$$f\left({\beta }_i,\beta \right)=\frac{b_2-\beta -{\beta }_i}{b_2-\beta -{\beta }_i+K_{r_{2}s}+\left(K_{r_{2}s}/K_{r_2}\right){\beta }_i},$$

$$g\left({\beta }_i,\beta \right)=\frac{{\beta }_i}{{\beta }_i+K_{r_{2}i}+\left(b_2-\beta -{\beta }_i\right)K_{r_{2}i}/K_{r_{2}s}}.$$

To describe crosstalk, we combine the models for the single receptors as follows. At the level of ligand-receptor activation the combined activation of EP2 and EP4 by PGE2 is modeled by

$$\begin{array}{l}\frac{dp}{dt}=-\left(1-\sigma \right){\mu }_1{r}_1{p}^{n_1}-\sigma {\mu }_2{r}_{2}p+{\delta }_1{r}_1^{\mathrm{*}}+{\delta }_2{r}_2^{\mathrm{*}},\\ \frac{d{r}_1}{dt}=-(1-\sigma ){\mu }_1{r}_1{p}^{n_1}+{\delta }_1{r}_1^{\mathrm{*}},& \frac{d{r}_1^{\mathrm{*}}}{dt}=(1-\sigma ){\mu }_1{r}_1{p}^{n_1}-{\delta }_1{r}_1^{\mathrm{*}},\\ \frac{d{r}_2}{dt}=-\sigma {\mu }_2{r}_{2}p+{\delta }_2{r}_2^{\mathrm{*}},& \frac{d{r}_2^{\mathrm{*}}}{dt}=\sigma {\mu }_2{r}_{2}p-{\delta }_2{r}_2^{\mathrm{*}}-{\rho }_2{r}_2^{\mathrm{*}},\end{array}$$ (14)

where $\sigma \in (0,1)$ indicates the fraction of PGE2 binding to EP4. This system is combined with the models for the G protein activation cycles for EP2 (7) and EP4 (10) and cAMP production (11) (with $\eta =1$) that take crosstalk between downstream effects in cAMP signaling into account,

$$\begin{array}{l}\frac{d{\alpha }_s^{\mathrm{*}}}{dt}=k_s{r}_2^{\mathrm{*}}{\sigma }_{0}f\left({\beta }_i,\beta \right)+k_s{r}_1^{\mathrm{*}}\left(1-{\sigma }_0\right)\frac{b_2-\beta -{\beta }_i}{b_2-\beta -{\beta }_i+K_{r_{1}s}}-d_s{\alpha }_s^{\mathrm{*}},\\ \frac{d{\alpha }_i^{\mathrm{*}}}{dt}=k_i{r}_2^{\mathrm{*}}g\left({\beta }_i,\beta \right)-d_i{\alpha }_i^{\mathrm{*}},\\ \frac{d{\beta }_i}{dt}=-k_i{r}_2^{\mathrm{*}}g\left({\beta }_i,\beta \right)+k_{\beta i}\beta \left(a_i-{\beta }_i-{\alpha }_i^{\mathrm{*}}\right)-d_{\beta i}{\beta }_i,\\ \frac{d\beta }{dt}=k_i{r}_2^{\mathrm{*}}g\left({\beta }_i,\beta \right)+k_s{r}_2^{\mathrm{*}}{\sigma }_{0}f\left({\beta }_i,\beta \right)+k_s{r}_1^{\mathrm{*}}\left(1-{\sigma }_0\right)\frac{b_2-\beta -{\beta }_i}{b_2-\beta -{\beta }_i+K_{r_{1}s}}+d_{\beta i}{\beta }_i\\ +d_{\beta s}\left(b_2-\beta -{\beta }_i\right)-\beta \left(k_{\beta s}\left(a_s-b_2+\beta +{\beta }_i-{\alpha }_s^{\mathrm{*}}\right)+k_{\beta i}\left(a_i-{\beta }_i-{\alpha }_i^{\mathrm{*}}\right)\right),\\ \frac{dw}{dt}=k_w\frac{{\alpha }_s^{\mathrm{*}}{A}^{tot}}{{\alpha }_s^{\mathrm{*}}+K_{as}\left(1+{\alpha }_i^{\mathrm{*}}/K_{ai}\right)}-d_w\frac{w{P}^{tot}}{w+K_w}.\end{array}$$ (15)

where ${\sigma }_0\in (0,1)$ indicates the fraction of ${\mathrm{G}}_{{\alpha }_s\beta \gamma }$-units binding to EP4.

4.2. Experimentally observed cAMP response

Figure 5 shows numerical simulations for the models (12)–(15). We only display cAMP levels for varying PGE2 concentrations to compare model predictions to the experimental data in Figures 3 and 4. Simulation results for the complete signaling pathways for EP2, EP4 and their crosstalk are provided in subsequent subsections where the models’ behavior and sensitivity to parameter variations is analyzed in greater detail.

Figure 5:

Simulated cAMP levels for the signaling pathways of EP2, EP4 and their crosstalk; models (12)–(15) with parameter values in Tables 3–5 for PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

Figure 5 shows that for the parameter values in Tables 3–5 the models (12) and (13) for the signaling pathways of EP2 and EP4 qualitatively predict the cAMP response for the individual receptors in Figures 3 and 4. EP2 shows the characteristic threshold behavior with respect to PGE2 while cAMP levels for EP4 gradually increase with increasing PGE2 concentrations and then decay over time. However, the maximum cAMP level for EP4 is higher than for EP2. Moreover, while the gradual increase of cAMP levels with respect to PGE2 for crosstalk is correct, the maximum cAMP levels predicted by model (14)–(15) for the combined EP2–EP4 signaling are higher than for the signaling of EP2 alone.

The total receptor concentrations $r_1^{\mathit{tot}}$ for EP2 and $r_2^{\mathit{tot}}$ for EP4 have an immediate impact on cAMP levels. Lowering $r_2^{\mathit{tot}}$ and assuming that $r_2^{\mathit{tot}}=r_1^{\mathit{tot}}$ the maximum cAMP levels for EP2 and EP4 coincide, as shown in Figure 6. As in Figure 5 we note that the models predict the qualitative cAMP response of the EP2 and EP4 correctly, but cAMP levels for the combined signaling of EP2 and EP4 are higher than for the signaling of the single receptors.

Figure 6:

Simulated cAMP levels for the signaling pathways of EP2, EP4 and their crosstalk; models (12)–(15) with parameter values in Tables 3–5, except $r_2^{tot}=4\mathrm{n}\mathrm{M}$, for PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

Due to lack of information about the specific AC and PDE isoforms in the cAMP signaling pathways of EP2 and EP4 we use the simplistic model (11) to describe cAMP regulation. It is very likely that relevant processes and/or the effects of additional enzymes that regulate subcellular cAMP levels are neglected. Even if the model is very simplistic, it suffices to modify $K_{ai}$, the dissociation constant for AC6 and ${\mathrm{G}}_{{\alpha }_i}$, and use a slightly lower value to reproduce the correct qualitative cAMP response displayed in Figures 3 and 4. Figure 7 shows that with the parameter values in Tables 3–5, except for $K_{ai}=0.017\mu M$, the models resemble the characteristic cAMP response for both, the single receptors and their crosstalk. The maximum cAMP levels for EP2 and EP4 coincide while the levels are lower for the combined signaling of the receptors. Hence, in the sequel we use $K_{ai}=0.017\mu M$ in the simulation studies, all other parameters are taken from Tables 3–5. Note that the parameter $K_{ai}$ is related to the G protein ${\mathrm{G}}_{{\alpha }_i}$ and hence, it only affects EP4 signaling and crosstalk but does not impact the EP2 signaling pathway.

Figure 7:

Simulated cAMP levels for the signaling pathways of EP2, EP4 and their crosstalk; models (12)–(15) with parameter values in Tables 3–5, except $K_{ai}=0.017\mu M$, for PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

4.3. EP2 signaling pathway

Figure 8 shows simulations for the full EP2 signaling cascade, i.e. system (12) with the parameter values in Tables 3–5. The cAMP levels resemble the characteristic threshold behavior of EP2 in Figure 3. For PGE2 concentrations of $0.01\mu M$ and $0.1\mu M$ there is nearly no cAMP production while cAMP levels stabilize quickly for the PGE2 concentrations $1\mu M$ and $10\mu M$.

We further observe that ligand-receptor binding dynamics is rapid. Hence, we compare the results with simulations where the quasi-steady state approximation (6) is used instead of the kinetic model (5) for ligand-receptor binding. The simulations are shown in Figure 9. The models behave nearly the same which justifies the quasi-steady state approximation. Both models predict the characteristic threshold behavior of EP2 in response to PGE2. However, in the sequel we include ligand-receptor binding dynamics since it is important for EP4 and for modeling crosstalk of the receptors.

The parameter values for the kinetic rate constants ${\mu }_1$ and ${\delta }_1$ for EP2 and PGE2 are not available in the literature. We therefore took the rate constant ${\delta }_1$ for the DcR3 decoy receptor in [20] and calculated ${\mu }_1$ given the equilibrium dissociation constant $K_D=12nM$ for PGE2 and EP2 in [15]. Figure 10 confirms that varying the kinetic rate constants while keeping their ratio ${\delta }_1/{\mu }_1$ equal to the equilibrium dissociation constant $K_D$ has very little effect on cAMP levels for EP2.

Figure 10:

Simulated cAMP levels for the EP2 signaling pathway; model (12) with parameter values in Tables 3–5 for varying kinetic rate constants ${\mu }_1$ and ${\delta }_1$ keeping their ratio equal to $K_D=12\mathrm{n}\mathrm{M}$, for PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

Finally, we analyze the impact of the Hill coefficient $n_1$ in the EP2 ligand binding dynamics on cAMP response. Higher order reactions are crucial to describe the experimentally observed switch between low and high cAMP levels dependent on PGE2 concentrations. Figure 11 shows cAMP levels predicted by model (12) with the parameter values in Tables 3–5 for varying Hill coefficients. For the Hill coefficients $n_1=1$ and $n_1=2$ cAMP levels increase gradually with increasing PGE2 concentrations while the threshold behavior is modeled by Hill coefficients $n_1\ge 3$.

Figure 11:

Simulated cAMP levels for the EP2 signaling pathway; model (12) with parameter values in Tables 3–5 for varying Hill coefficients, for PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

4.4. EP4 signaling pathway

Figure 12 shows simulations for the full EP4 signaling cascade, i.e. model (13) with the parameter values in Tables 3–5, except $K_{ai}=0.017\mu M$. The cAMP levels resemble the characteristic signaling response of EP4 in Figure 3. The levels increase gradually with increasing PGE2 concentrations and show a sharp increase and then decay over time.

Figure 12:

Simulation of the EP4 signaling pathway; model (13) with parameter values in Tables 3–5, except $K_{ai}=0.017\mu M$, for PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

Different from EP2 where ligand-receptor binding is rapid and the quasi-steady state assumption (6) is justified, explicit modeling of ligand-receptor binding dynamics plays an important role for EP4 since the receptor internalizes rapidly. The kinetic model for ligand-receptor interactions (13) provides a natural way to include internalization and the internalization rate ${\rho }_2$ leads to the experimentally observed decay in cAMP levels over time. Figure 13 illustrates the effect of varying ${\rho }_2$ on the cAMP response of EP4. For ${\rho }_2=0$ the dynamics resembles the results for the EP4 signaling pathway with the quasi-steady state approximation, as in (6) but with $r_2^{\mathrm{*}}$ and $r_2^{\mathit{tot}}$ and the corresponding parameters, see Figure S1 in Supplementary Material. We note that altering the internalization rate changes the decay properties but has little impact on maximum cAMP levels.

Figure 13:

Simulated cAMP levels for the EP4 signaling pathway; model (13) with parameter values in Tables 3–5, except $K_{ai}=0.017\mu M$, for varying ${\rho }_2$ and PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

EP4 activates the stimulating G protein ${\mathrm{G}}_{{\alpha }_s}$ and the inhibiting G protein ${\mathrm{G}}_{{\alpha }_i}$. In model (13) the effects of competitive binding of ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$ to the receptor are determined by the ratio $K_{r_{2}s}/K_{r_{2}i}$ of the EP4-dissociation rates for ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$. Figure 14 shows the impact of competitive binding on the EP4 signaling pathway where cAMP levels for different values of $K_{r_{2}i}$ are shown. Higher values for $K_{r_{2}i}$ lead to higher cAMP levels. Similarly, varying $K_{r_{2}s}$ has an opposite effect on cAMP response, i.e. cAMP levels decrease with increasing $K_{r_{2}s}$, see Figure S2 in Supplementary Material.

Figure 14:

Simulated cAMP levels for the EP4 signaling pathway; model (13) with parameter values in Tables 3–5, except $K_{ai}=0.017\mu M$, for varying $K_{r_{2}i}$ and PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

Competitive binding also plays an important role for cAMP expression since $G_{{\alpha }_i}$ prevents binding of AC to $G_{{\alpha }_s}$. In model (13) competitive binding of $G_{{\alpha }_s}$ and $G_{{\alpha }_i}$ to AC is determined by the parameter $K_{ai}$. Varying the dissociation constant $K_{ai}$ for AC6 and ${\mathrm{G}}_{{\alpha }_i}$ alters cAMP levels but does not change the qualitative cAMP dynamics nor the gradual increase of cAMP with respect to PGE2, see Figure S3 in Supplementary Material.

4.5. EP2-EP4 crosstalk

Figure 15 shows simulations for the combined signaling pathways of EP2 and EP4, i.e. model (14)–(15) with the parameter values in Tables 3–5, except $K_{ai}=0.017\mu M$. The cAMP levels resemble the characteristic signaling response for receptor crosstalk in Figure 3. As for EP4 the cAMP maximum levels increase gradually with increasing PGE2 concentrations and the transient nature of the response remains. However, the maximum cAMP levels are lower than for the signaling of EP2 or EP4 alone, see also Figure 7.

Figure 15:

Simulation of the combined EP2-EP4 signaling pathway; model (14)–(15) with parameter values in Tables 3–5, except $K_{ai}=0.017\mu M$, for PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

The impact of varying the EP4 internalization rate ${\rho }_2$ on EP2-EP4 crosstalk is similar as for EP4. Increasing ${\rho }_2$ leads to a larger decay in cAMP concentration over time and for ${\rho }_2=0$ the dynamics resembles the model without ligand dynamics, i.e. replacing the equations for ligand-binding dynamics by the quasi-steady state approximation, see Figure S4 in Supplementary Material. The effects of competitive binding in the G protein activation cycles of EP4 are determined by the ratio $K_{r_{2}s}/K_{r_{2}i}$ of the dissociation rates for ${\mathrm{G}}_{{\alpha }_i}$ and ${\mathrm{G}}_{{\alpha }_s}$. Varying the dissociation rate $K_{r_{2}i}$ has the same qualitative effect on cAMP levels for crosstalk as for EP4 alone as shown in Figure 14 (see also Figure S5 in Supplementary Material). The inhibiting effect of ${\mathrm{G}}_{{\alpha }_i}$ on cAMP production is modeled by competitive binding between ${\mathrm{G}}_{{\alpha }_i}$ and ${\mathrm{G}}_{{\alpha }_s}$ to AC and determined by the dissociation constant $K_{ai}$ for AC6 and ${\mathrm{G}}_{{\alpha }_i}$ in (15). Using a slightly lower value, $K_{ai}=0.017\mu M$, than in Table 5 was necessary to obtain the correct cAMP response for crosstalk, see Figure 5. Increasing $K_{ai}$ leads to higher cAMP levels for crosstalk but does not change the qualitative cAMP dynamics in response to PGE2, see Figure S6 in Supplementary Material. Similar to EP4 signaling, cAMP levels decrease with increasing $K_{r_{2}s}$, see Figure S7 in Supplementary Material.

Finally, we analyze the effects of the crosstalk binding ratios $\sigma ,{\sigma }_0\in (0,1)$ in model (14)–(15). Figure 16 shows cAMP levels when varying $\sigma$. The binding ratio $\sigma$ indicates the fraction of PGE2 binding to EP4 while $1-\sigma$ is the fraction of PGE2 binding to EP2. For $\sigma =0$ the cAMP levels resemble the results for EP2 in Figure 7 while for $\sigma =1$ the cAMP levels resemble the results for EP4 in Figure 7. Note that cAMP levels for $\sigma =0$ and $\sigma =1$ are lower than for the single receptors, see Figure 7, due to the binding ratio ${\sigma }_0=0.5$ in the G protein activation cycles indicating that half of the ${\mathrm{G}}_{{\alpha }_s}$ protein complexes bind to EP2 and half to EP4.

Figure 16:

Simulated cAMP levels for the combined EP2-EP4 signaling pathway; model (14)–(15) with parameter values in Tables 3–5, except $K_{ai}=0.017\mu M$, for varying $\sigma$ and PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

Figure 17 shows cAMP levels when varying the binding ratio ${\sigma }_0$ for EP2 and EP4 and the G protein ${\mathrm{G}}_{{\alpha }_s}$. Small values of ${\sigma }_0$ reflect that ${\mathrm{G}}_{{\alpha }_s}$ binds with high probability to EP2 and with low probability to EP4. For ${\sigma }_0=0$ the cAMP levels for crosstalk resemble the results for EP2 signaling while for ${\sigma }_0=1$ the results for EP4 signaling, see Figure 7. Simultaneous variations in $\sigma$ and ${\sigma }_0$, with $\sigma ={\sigma }_0$, affect the qualitative cAMP dynamics, where the cAMP levels for lower values resemble the results for EP2 and for higher values the results for EP4 signaling, but have smaller impact on the maximum levels of cAMP when $\sigma ={\sigma }_0=0$ and $\sigma ={\sigma }_0=1$, see Figure S8 in Supplementary Material.

Figure 17:

Simulated cAMP levels for the combined EP2-EP4 signaling pathway; model (14)–(15) with parameter values in Tables 3–5, except $K_{ai}=0.017\mu M$, for varying ${\sigma }_0$ and PGE2 concentrations $0.01\mu \mathrm{M},0.1\mu \mathrm{M},1\mu \mathrm{M},10\mu \mathrm{M}$.

5. Conclusion and discussion

We propose mathematical models for the cAMP signaling cascades of the GPCRs EP2 and EP4 and their combined signaling. The models qualitatively predict the characteristic cAMP profiles in response to varying PGE2 concentrations observed experimentally [2] and help to identify processes in the complex signaling pathways that can explain the differences in the signaling behavior of the receptors and their crosstalk. While many models assume ligand-receptor binding is rapid and make quasi-steady state assumptions, it is becoming increasingly important to incorporate specific ligand-receptor binding kinetics [3, 8]. We include a dynamic model for ligand-receptor binding where we take EP2 and EP4 specific ligand binding dynamics into account such as ultrasensitivity of EP2 and internalization for EP4. On the one hand, this allows us to explain the qualitative differences in the cAMP profiles of EP2 and EP4. On the other hand, we can easily combine the models to describe crosstalk at the level of ligand-receptor activation. More refined modeling approaches that describe receptor activation as a separate step and allow inactive receptors to bind to G proteins, see e.g. [21], [22], is subject to future research. While most models for GPCRs assume that a single G protein is activated [4, 5, 8, 9] only few describe the activation of multiple pathways through the coupling to different G proteins [27, 28], as considered here. We propose a novel model for EP4, a GPCR that activates two different G proteins and incorporate the effects of competitive binding in the activation cycles for ${\mathrm{G}}_{{\alpha }_s}$ and ${\mathrm{G}}_{{\alpha }_i}$.

In the ligand-receptor binding kinetics ultrasensitivity of EP2 with respect to PGE2 is modeled by a Hill coefficient $n_1=3$ leading to the threshold behavior displayed in Figure 3. A possible explanation for ultrasensitivity is dimerization. If binding of PGE2 to EP2 is followed by the formation of dimers this can increase the probability that another receptor starts signaling and hence, leads to positive cooperativity [24]. It was shown that dimerization can dominate over the canonical allosteric mechanism for generating cooperativity [24], i.e. it can result in positive cooperativity and thus, in a Hill coefficient greater than one. However, while dimerization is documented for other GPCRs it remains to be identified for EP2 [2]. Here we choose the Hill coefficient $n_1=3$ since Hill coefficients greater than three are considered very large when modeling dimerization. Another possible explanation for the threshold behavior in EP2 signaling is bistability [11], i.e. the system can switch between two distinct stable steady states [12]. Bistability can be approximated by a large Hill coefficient $n_1$, but further experimental data is required to determine whether EP2 shows bistability or ultrasensitivity. For EP4, the cAMP signaling response to PGE2 is significantly different. In this case, cAMP expression increases gradually with PGE2 levels and we take as Hill coefficient $n_2=1$. In contrast to EP2 that does not internalize, internalization plays a key role in EP4-induced cAMP levels. Internalization is taken into account in the kinetic model for ligand-EP4 binding leading to a peak and then decay of cAMP levels over time as displayed in Figure 3.

Due to too many unknown aspects concerning the specific AC and PDE isoforms, and possibly additional processes involved in intracellular cAMP regulation, we use a simplistic model for cAMP production assuming that AC6 and PDE4 are the dominant isoforms regulating cAMP levels. However, other isoforms can be involved, instead of, or in addition to, AC6 and PDE4. The regulation patterns for AC isoforms differ significantly, see Table 3, and some isoforms are also inhibited or stimulated by the G protein ${\mathrm{G}}_{\beta \gamma }$ which we do not consider here. To develop an improved, more detailed model requires additional knowledge on the specific AC and PDE isoforms. Our model provides a basis and framework for future studies as it allows to identify key parameters responsible for cAMP regulation in the EP2 and EP4 signaling pathways. The simulation results illustrate how changes in parameter values affect the cAMP response of the receptors and their crosstalk. One could examine the effects of different AC isoforms on cAMP levels by modifying parameter values and/or the equation for cAMP production (11).

Many aspects in the cAMP signaling cascades of EP2 and EP4 are still unknown. Moreover, few precise parameter values are available for the specific cell types and GPCRs used for the experimental data displayed in Figure 3. Some of the parameter values in the simulations stem from different organisms, cell types and/or receptors and typically, values vary significantly depending on the cell type, organism and experimental setup. Here, we balance between detailed modeling of the important steps in the cAMP signaling pathways of EP2 and EP4 and inserting uncertainties due to unknown parameter values. Note that the experimental data indicates only relative cAMP levels, so it does not provide information on the magnitude of intracellular cAMP levels. Therefore, our models focus on qualitative predictions and do not provide quantitative outcomes.

Increased PGE2 concentrations are observed in the tumor microenvironment of several cancer types [42]. Since PGE2 regulates immune cell function, the selective modulation of EP receptor signaling pathways has been shown to improve antitumor immune responses [43]. Better insight into the EP2 and EP4 signaling behaviors is crucial to efficiently control the cellular responses to PGE2. The mathematical model developed here could contribute to the design of more effective receptor-specific antagonists to be used as anticancer therapies.

Table 4:

Parameter values for the G protein activation cycles

symb.	parameter	value	reference
$k_s$	$G_{{\alpha }_s}$ activation rate	1–5s−1 (5s−1)	[9] (${\alpha }_{2a}$-adrenergic recept.)
$k_i$	$G_{{\alpha }_i}$ activation rate	5s−1	[9] (${\alpha }_{2a}$ adreno recept.)
$d_s$	$G_{{\alpha }_S^{GTP}}$ hydrolysis rate	0.04, 0.07s−1(0.07s−1)	[9]
$d_i$	$G_{{\alpha }_i^{GTP}}$ hydrolysis rate	0.03s−1	[9]
$k_{\beta s}$	$G_{{\alpha }_s^{GDP}}-\beta \gamma$ assoc. rate	0.7μM−1s−1	[9] (estimate)
$k_{\beta i}$	$G_{{\alpha }_i^{GDP}}-\beta \gamma$ assoc. rate	0.7μM−1s−1	[9] (rat, bovine)
$d_{\beta s}$	${\mathrm{G}}_{{\alpha }_s^{GDP}\beta \gamma }$ dissoc. rate	18.9 · 10−3s−1	[9] (estimate)
$d_{\beta i}$	${\mathrm{G}}_{{\alpha }_i^{GDP}\beta \gamma }$ dissoc. rate	0.14 · 10−3s−1	[9] (estimate)
$K_{r_{1}s}$	EP2-${\mathrm{G}}_{{\alpha }_s^{GTP}\beta \gamma }$ dissoc. const.	0.8μM	[9] ($\beta$-adrenergic recept.)
$K_{r_{2}s}$	EP4-${\mathrm{G}}_{{\alpha }_s^{GTP}\beta \gamma }$ dissoc. const.	0.088μM	[38] (human)
$K_{r_{2}i}$	EP4-${\mathrm{G}}_{{\alpha }_i^{GTP}\beta \gamma }$ dissoc. const.	50, 175, 214nM (214nM)	[38] (human)
$a_s$	total $G_{{\alpha }_s}$ concentration	2.3μM	[9] (myocyte)
$a_i$	total $G_{{\alpha }_i}$ concentration	8μM	[9] (neutrophils)
$b_1$	total $G_{\beta \gamma }$ concentr. EP2	0.005μM	[9] (neutrophils, estim.)
$b_2$	total $G_{\beta \gamma }$ concentr. EP4	1.8μM	[9] (estimate)
${\sigma }_0$	binding ratio for crosstalk	0.5/0.7	assumed