K⁺ channels.

We have included the ATP-sensitive potassium (K_ATP) channel and the voltage gated K⁺ (K⁺_Kr) channel. Messengers can affect K⁺ channel dynamics in various ways. We have taken into account that protein kinase A (PKA) activation and a decrease in phosphatidylinositol 4,5-bisphosphate (PIP₂) concentration can lead to block of K_ATP channels [38, 41, 42] (see CM for details). Several studies suggest that voltage-gated K⁺ channels might be also regulated by PIP₂ [43, 44]. However, K_v2.1 channels, that are the main voltage-gated K⁺ channels in β-cells [2, 4] are not sensitive to PIP₂ depletion [43]. For this reason we excluded the influence of PIP₂ on voltage-gated K⁺ channels.

Ca²⁺ channels and pumps.

PM depolarization potentiates glucose-induced Ca²⁺ influx through voltage-gated Ca²⁺ channel (VDCCs). Depletion of intracellular Ca²⁺ stores in β-cells activates a special Ca²⁺-release activated current (or store operating current (SOC)). While the detailed mechanism for coupling of ER Ca²⁺ store depletion with these PM channels activation is uncertain, it likely involves translocation of STIM1 and STIM2 proteins from the ER calcium stores to the PM where they interact with ORAI1 and related proteins associated with cation influx channels [45, 46]. Plasma membrane Ca²⁺ pumps provide an outward Ca²⁺ current.

Na⁺ channels.

Voltage-gated Na⁺ current is likely inactivated in mouse β-cells [47] due to the relatively high resting V_p, so it is not considered, although it may play a role in human beta cells (see [30]). Nonselective PM cation channels permeable to Na⁺ and Ca²⁺ have been described in pancreatic β-cells, and the transient receptor potential ion channels were proposed for this role [48, 49]. Specific nonselective cation channels (NALCN) are expressed in pancreatic islets and can also be modulated by acetylcholine through activation of M₃ muscarinic receptors [39, 50]. We modeled these currents as Na⁺ background current (I_Nab) through specific channels on PM (see CM).

Incretin hormones and receptors in β-cells.

Glucagon-like-peptide-1 (GLP-1) is one potent incretin hormone coupled primarily to the specific α_s-G-protein-coupled receptor (GLP-1R). The GLP-1R belongs to the class B family of GPCRs. The GLP-1R couples to the G-protein complex and facilitates the release of the activated Gα_s subunit of the complex, which activates plasma membrane-bound adenylyl cyclase (AC_P) to produce cAMP [12, 63, 64]. Following agonist-induced receptor activation, GLP-1R is probably internalized [65]. Small ubiquitin-related modifier protein (SUMO) can covalently modify GLP-1R and thereby desensitize GLP-1 signaling [56].

In islets, GLP-1R is expressed predominantly on β-cells [66]. A mathematical model for GLP-1R signal transduction was published recently [24]. However, this model lacks components such as a complex of G-protein with AC_p. To address this we used our general model for GPCR effects (for details see CM).

Another important incretin hormone is glucose-dependent insulinotropic polypeptide or gastric inhibitory polypeptide (GIP). GIP is a 42-amino-acid hormone, secreted from the enteroendocrine K cells and one of the major mediators in the regulation of nutrient-dependent insulin release from the pancreas [67]. Hyperglycemic clamp studies or GIP infusion established that GIP was insulinotropic at physiological concentrations [68]. GIP exerts its effects through binding to a specific α_s-G-protein-coupled receptor (GIPR) that also belongs to the B-family class of GPCRs of the glucagon–secretin family of peptides [68, 69].

The GIPR undergoes rapid and reversible homologous desensitization following binding of GIP [68, 70]. Regulator of G-protein signaling-2, G-protein receptor kinase 2, and β-arrestin 1 all have been implicated in GIPR desensitization [71].

Interestingly, class B GPCRs are characterized by a common topology of the ligand-receptor complex, and by their ability to couple multiple G-proteins. Biochemical and structural studies have led to a model of class B1 GPCR activation by peptide hormones including GLP-1 and GIP, referred to as the “two-step” mechanism [72]. This corresponds to collision coupling, where an agonist binds to the free receptor and then the agonist-receptor complex ‘‘collides” with the free G-protein. Corresponding models were developed for GLP-1R and GIPR (see CM).

Adrenergic receptors.

Adrenergic agonists, the endogenous catecholamines adrenaline (epinephrine) and noradrenaline (norepinephrine), as well as agonists such as clonidine, all inhibit insulin secretion [10, 73, 74]. G-protein-coupled α_2A-adrenergic receptors (AdR) are responsible for these effects for adrenaline/noradrenaline, and they are mediated by the pertussis toxin-sensitive heterotrimeric G_i and G_o proteins, that inhibit AC [10, 73]. The mRNAs encoding the α_2A adrenoceptor are expressed at relatively high levels in human islets [5].

Co-immunoprecipitation studies showed pre-coupling of a_2A-adrenergic receptors with G_i/o proteins [75]. However, no evidence for a_2A-adrenergic receptor and G_i-protein precoupling was found in resonance energy transfer studies and it was suggested that a_2A-adrenergic receptors and G-proteins interact by rapid collision coupling [76]. For this reason we also used the collision coupling mechanism for α_2A-adrenergic receptor (see CM). Desensitization mechanisms have not been specifically studied in pancreatic β-cells for these receptors, however, these receptors recycle rapidly in COS-7 and HEK-293 cells [77].

We modeled the processes of desensitization of GLP-1R, GIPR and AdR as a simplified mechanism that includes only one step for the active state (where a receptor is bound with ligand) pass off or recirculation of GPCRs (see CM), because these mechanisms are not well studied.

Adenylyl cyclases and phosphodiesterases.

Activated adenylcyclase (AC) synthesizes cAMP from the substrate Mg²⁺ATP. Rodent and human β-cells and insulinoma cell lines express different AC isoforms, including the Ca²⁺ and calmodulin-activated isoforms AC1, AC3, and AC8 [78–81]. Interestingly, among the cloned and characterized members of the mammalian AC family, two isoforms, AC1 and AC8, are synergistically activated by Ca²⁺-calmodulin and G_sα [82, 83]. Both receptors for GIP and GLP-1 activate the α subunit (G_sα) of the G-protein leading to an activation of AC in pancreatic cell lines and isolated pancreatic islets [67, 84]. The activation of AC by the α subunit (G_sα) of the G-protein may be associated with formation of their complex with PM bound AC (AC_p) [85, 86].

GLP-1 resulting in increments in G_sa has been proposed to regulate calmodulin-activated AC8 that is central to GLP-1 signaling in rodent and human β-cells [78, 79, 87]. Our previous kinetic analysis supports this conclusion [23]. Similar interactions with AC8 was suggested for GIPR [68]. Catecholamines inhibit AC, however it is not known which isoform of AC in the β-cell is inhibited [10]. Activation of G_i by catecholamines may block AC_P mediated cAMP synthesis thus preventing the augmentation of insulin release stimulated by GLP-1 and GIP [10]. Acute activation of G_i/o-coupled receptors leads to inhibition of AC8 isoform [88]. Therefore we suggest that this G_i binds to the same AC isoform as G-proteins activated by GLP-1 or GIP, but AC bound with G_i has no catalytic activity. We found that these mechanisms of interaction between G-proteins and target enzymes (represented for AC_p in Fig 2) can play an important role in the regulation of messenger pathways (see Results and discussion).

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Fig 2. A schematic model of interaction of G-proteins activated by receptors with PM bound adenylyl cyclase (AC_p).

Activating G-protein can bind with AC_p and their complex can accelerate cAMP production. AC_p complex bounded with inhibiting G-protein cannot catalyze cAMP production.

https://doi.org/10.1371/journal.pone.0152869.g002

Some soluble cytoplasmic adenylate cyclases (AC_c) can also contribute to cAMP synthesis that are not controlled by G-proteins and forskolin but rather by calcium, ATP and bicarbonate [89]. Therefore alongside G-protein activated or inhibited AC_p we modeled a cytoplasmic AC_c that is insensitive to G-protein and can be activated by glucose and/or Ca²⁺ (see for detals CM).

Some resting cAMP concentration exists in the absence of agonists (see below). Mechanisms of AC activation were not studied in this case for β–cells. We suggest that AC activity in this case can be a consequence of basal activity of AC_c and constitutive G-protein dependent receptor for AC_P that is observed in other GPCRs.

Phosphodiesterases (PDEs) hydrolyze cAMP and cyclic GMP. There are eleven known PDE families that differ in primary structures, responses to specific effectors and sensitivities to specific inhibitors, cellular expression and intracellular location and may modulate distinct regulatory pathways within the cell [90, 91]. In β-cells, it has been suggested that several PDE isoforms (1C, 3B, 4, 8B, and 10A) are involved in regulation of insulin secretion [91, 92]. In our model we consider both Ca²⁺ bound calmodulin activated and constitutive PDE activity as in a previous model [23] (see CM).

PKA and Epac.

The two primary downstream effectors of cAMP are PKA and the cAMP-regulated guanine nucleotide exchange factors (Epac) that play a critical role in insulin release [64, 93]. PKA is a holoenzyme composed of catalytic and regulatory subunits. The catalytic subunits release when cAMP binds to the regulatory subunits and then act to phosphorylate downstream substrates. Epac is similarly activated by cAMP [64, 93]. We have used a simple mathematical model of these events [25]. There are multiple phosphorylation targets of PKA and Epac that are related to their effects on exocytosis. PKA and Epac pathways can have different physiological relevance based on differences in cAMP binding constants and downstream targets [94]. However, in the present model, downstream targets of PKA and Epac include only PKA blocking of K_ATP channels (see Eq 13, CM) and regulation of the IP₃ receptors in the ER (see CM).

PLC pathway receptors.

Pancreatic β cells express various G_q-coupled receptors, including the muscarinic receptors, FFAR1/GPR40, GPR120 and different P2Y receptor subtypes, that can regulate PLC activity [67]. However, here we included only what seem to be the most important for the β-cell: muscarinic acetylcholine receptor (MR) and FFAR1/GPR40. Others of this class seem to be expressed at lower levels [5, 104].

Muscarinic acetylcholine receptor (MR).

Acetylcholine binds to MRs in the β-cell membrane coupled to Gα_q/11 to activate PLC [3]. mRNA encoding the type 3 MR is the most abundant in human islets but other isofroms are expressed in β-cells [5].

Initially, it was thought that MR and G-proteins interact freely with each other on the cell membrane [105]. However, type 3 MRs can pre-couple with their preferential classes of G-proteins [51] and here we used a pre-coupled model (see CM). Agonist-bound MR are phosphorylated by G-protein-coupled receptor kinases that leads to receptor internalization and recycling (or down regulation) [55].

Free fatty acid receptor (FFAR1/GPR40).

The class A G-protein-coupled receptor GPR40, now also known as free fatty acid receptor 1 (FFAR1) is predominantly expressed in mouse, rat and human pancreatic β-cells and plays a major part in fatty acid amplification of GSIS [106]. FFAR1/GPR40 is activated by medium- to long-chain fatty acids and it is predominantly coupled to Gα_q that typically signals through PLC-mediated hydrolysis of membrane phospholipids and is coupled to the formation of IP₃ and DAG [5, 106]. A collision coupling mechanism was suggested as the predominant mechanism for FFAR1/GPR40 (see [107]). Interestingly, investigations of β cell-specific inactivation of the genes encoding the Gα_q/11-coupled receptor have shown that glucose does not directly activate this receptor [108]. In stably transfected HEK-293 cells, FFAR1/GPR40 receptors underwent rapid agonist-induced internalization [109]. However, in the absence of specific data we used a general mechaism for modeling the processes of desensitization for MR and FFAR1/GPR40 in β-cells (see CM).

Phospholipase C (PLC).

At least thirteen different PLC family members have been identified in various mammalian tissues and the consensus is that PIP₂ is the major substrate of PLC yielding DAG and IP₃ [110, 111]. Mouse and rat islets have been reported to express members of the three major phospholipase (PLC) subfamilies; PLC-β1,—β2,—β3,—β4, PLC-γ1, PLC-δ1 and PLC-δ2 [112, 113]. These enzymes appear to be activated by different mechanisms. It is likely that PLC-β isoforms, activated by carbachol, are localized on the PM and are regulated by the Gα_q subfamily of G-proteins [114]. On other hand PLCδ1 is localized in the cytoplasm and regulated by nutrients including glucose [115]. PLC-γ1 is entirely cytoplasmic in MIN6 cells [116]. Interestingly, PLC-γ is directly phosphorylated and activated by tyrosine kinase [114]. Therefore we modeled the two main PLC forms in β-cells as a nutrient (glucose) activated cytoplasmic PLC form (PLC_C) and as a G-protein-coupled PLC that is activated by specific G_q-type G-proteins in the PM (PLC_P) (Fig 1).

PLC- β and Gα_q can form a complex mediating PLC activity [114]. We therefore suggested that the PLC_P can be activated only when this G-protein binds with its corresponding PLC (see CM). Interestingly, all the PLC isoforms discussed here can be activated by Ca²⁺ [110, 111]. However, the cytoplasmic PLC isoform activated by glucose is more strongly controlled by Ca²⁺ availability than the isoform that is activated by agonists [115, 117]. We thus considered the constant for [Ca²⁺]_c activation for PLC_p to be higher (0.4 μM) in comparison with the PLC_C (0.2 μM) (see CM).

IP₃ and DAG handling.

The dynamic intracellular concentrations of IP₃ and DAG are determined by the rates of synthesis and degradation. We used our simple model [33] for modeling IP₃ synthesis and degradation. The DAG model was developed similarly to the IP₃ model (see CM). However, DAG remains bound within the PM and it was simulated as a concentration closely associated with the PM.

Regulation of PKC activation.

DAG promotes membrane recruitment and activation of protein kinase C (PKC), an enzyme linked to the regulation of many cellular processes [118]. Several PKC isoenzymes are expressed in pancreatic β-cells and it is known that PKC phosphorylates and activates the components of the exocytotic machinery in β-cell that is important for the regulation of insulin secretion. However, the precise role of PKC-family proteins in β-cell physiology is still controversial and the phosphorylation targets that mediate the enhancement of exocytosis by PKC are not well known [119]. For this reason in our quantitative model of messenger interactions we consider only activation of P4P synthesis from PI by activated PKC (see Fig 1).

DAG can also activate PDK1 in mouse islets [120] and this kinase is also responsible for stimulating insulin release [121]. However, activated PDK1 does not seem to directly regulate the processes that we consider here and its regulation is not further considered.

Regulation of Ca²⁺ dynamics.

Calcium is a ubiquitous second messenger that regulates many biological processes, such as cell signaling and transport, membrane excitability and substance secretion. In the β-cell Ca²⁺ homeostasis is controlled by transmembrane ion flux and involves coordinated interplay between multiple ion transport mechanisms and organelles. Ca²⁺ enters the β-cells primarily through voltage-gated Ca²⁺ channels. Additionally, Ca²⁺ store-operated current (SOC) increases [Ca²]_c (see CM). The plasma membrane of β-cells contains P-type ATPases that pump Ca²⁺ out of the cytosol to the extracellular environment [122, 123].

Ca²⁺ enters the endoplasmic reticulum (ER) via P-type ATPases (SERCA pumps, primarily SERCA2 and 3 in β-cells) using ATP and leaves predominantly through intracellular ER Ca²⁺ channels. The type III inositol 1,4,5 triphosphate receptor (IP₃R) is the main calcium release channel in the β-cell ER. When IP₃ binds to IP₃R, Ca²⁺ passes from the lumen of ER into the cytosol [3, 122–124]. An additional regulatory control of IP₃R is possible through phosphorylation of IP₃R or through the associated proteins that bind to the regulatory domain. PKA activation leads to an opening of IP₃R [125–127] (Fig 1). We have included models of IP₃R in our general model (see CM).

Incretin hormones.

GIP is secreted in the same time and concentration frame as GLP-1 and these concentrations are sufficient to activate GLP-1R or GIPR in humans [141, 142]. GLP-1 and GIP receptor activation on β-cells leads to increased cAMP and intracellular Ca²⁺ that result in exocytosis of insulin-granules as a late response to oral glucose (20–120 min) [143]. GLP-1 and GIP have insulinotropic action in healthy humans [141, 144, 145].

We modeled the actions of incretin hormones and the resulting activation of AC_P that increases cAMP synthesis and PKA activation. To do this we set up a simulation with GLP-1 and GIP levels 20 times higher than the Michaelis constant for GLP-1 or GIP receptor binding (coefficient K_GLP1, or K_GIP, Eqs 46 and 47). As illustrated in Figs 1 and 3 simulation of increased cAMP leads to dual activation of the PKA and Epac branches of the cAMP pathway. Indeed it was found that GLP-1 and GIP lead to additional glucose-dependent closure of K_ATP channels and an increase in Ca²⁺ release from ER leading to [Ca²⁺]_c increase, that accelerates Ca²⁺ dependent insulin secretion [64, 68, 93, 146]. We were able to simulate increased [Ca²⁺]_c (and a corresponding fall in [Ca²⁺]_ER) by GLP-1R activation as a consequence of IP₃R stimulation through PKA activation (Fig 3).

However, in low glucose concentrations treatment of pancreatic β-cells with agents that increase cAMP- alone has little or no effect on insulin secretion. Only a combination of GLP-1 or GIP and high glucose potentiates GSIS [84, 147]. This important feature reduces the chance of producing hypoglycemia in patients with T2D with GLP-1 agonists [148]. Our computational model agrees with this behavior (see Fig 5, left part). At low glucose when [Ca²⁺]_c is low, activation of GLP-1R significantly increases AC_P (Fig 5B), however this does not lead to increased cAMP because increased [Ca²⁺]_c is required for ACp activation. On the other hand the model predicts that a facilitation of cAMP production by AC_P by GLP-1R agonists can be stimulated if glucose increases the concentration of Ca²⁺-dependent calmodulin that can activate PM bound AC in β-cells (see Figs 3 and 5). A small effect of GLP-1 on increased cAMP at low glucose was also simulated in our previous model of AC activation and of cAMP dynamics [23] becouse this effect can be explained by the sensitivity of AC_P to [Ca²⁺]_c changes rather than by the properties of GLP-1R. (A similar simulation of GLP-1 effects via AC activation was reported [24]). This and our previous analysis [23] provides support for a pivotal role of Ca²⁺/CaM-dependent AC activation (possibly the AC8 isoform) in GLP-1 effects. The potentiating effect of cAMP on insulin secretion requires synergism between the cAMP pathway and the Ca²⁺ signal. Our model of cAMP dynamic regulation may also help to explain the mechanism responsible for the pattern of cAMP changes that take place during phases changes and oscillations in intracellular Ca²⁺ [23].

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Fig 5. Simulation of GLP-1 and GIP action at low and high glucose levels.

(A) Cytoplasmic [Ca²⁺]_c and cAMP dynamics. (B) AC_GLP, AC_GIP and AC_AR are the concentrations of complexes of AC on PM (AC_P) bound with corresponding G-proteins activated by GLP-1R, GIPR and α_2A adrenergic receptors. (C) V_ACp is G-protein and Ca²⁺ dependent AC activity on PM, V_ACc is the glucose and Ca²⁺-activated soluble AC activity that is independent on G-protein. At low glucose concentrations (3 mM) (left part) GLP-1 was increased from basal level (3.1e-7 μM) to 6.2e-4 μM at 1000 sec. Increased extracellular glucose level was simulated at 3000 sec (from 3 mM to 8 mM). Than GIP administration was modeled as the increased GIP from basal level (1.37e-6 μM) to 3.42e-3 μM at 5000 sec.

https://doi.org/10.1371/journal.pone.0152869.g005

This model also provides an explanation for the observed clinical interaction between GLP-1R agonists and K_ATP channel blockers. Several human clinical trials have demonstrated that patients using GLP-1R agonists and K_ATP channel blockers therapies have a greater incidence of hypoglycaemia than those not using K_ATP channel blockers [148]. K_ATP channel blockers have also been shown to abrogate the glucose dependence of GLP-1R agonist activity in the rat pancreas [149].

Although the mechanism of this proposed uncoupling event has not been fully elucidated experimentally, our simulation suggests that K_ATP channel blockers may lead to modest PM depolarization and increased Ca²⁺ concentration even in low glucose (not shown) leading to activation of Ca²⁺ dependent AC on PM and an increased cAMP (and corresponding insulin secretion) in the presence of GLP-1R agonists. Thus, block of K_ATP channels independently of glucose may allow GLP-1R agonists to bypass their inherent glucose requirement by stimulating the downstream effects ordinarily associated with increased glucose. Our simulation shows also that similar behavior may also occur for GIPR activators in the presence of K_ATP channel blockers (not shown).

GLP-1 and GIP interaction.

In rodent and healthy human subjects the insulinotropic effectiveness of GLP-1 and GIP may be additive [141]. We were able to simulate an additional increase in cAMP concentration following GIPR activation with GLP-1 administration (Fig 5B). This synergetic GIP effect takes place because GLP-1R cannot fully activate AC_p and GIPR administration leads to additional AC_p activation in our model, i.e. a sum of AC_p activated by GLP-1R (AC_GLP) and GIPR (AC_GIP) was higher than AC_GLP with GLP-1 activation only.

However, investigations in mice have shown a less protracted action of GIP vs GLP-1 that would suggest a more prolonged activation of the GLP-1 receptor than that of GIPR [150]. One hypothesis, is that GIP provokes accelerated desensitization of the GIPR [150]. Ligand-induced desensitization and trafficking of GPCRs have been implicated as critical mechanisms for modulating response duration in vivo [151]. A naturally occurring variant of the GIP receptor underwent enhanced agonist-induced desensitization in adipocytes, which impaired GIP control of adipose insulin sensitivity [152]. We have exploited our mathematical model to examine the influence of receptor desensitization in cAMP dynamics. According to results of our modeling an increase in the rate constant for desensitization and/or a decrease of rate of return to surface for GIPR can actually decrease AC_p activity bound with GIPR to compare with AC bound with GLP-1R (not shown). However, we were not able to find any data where a signaling desensitization in response to these two hormones has been compared under the same experimental setting in pancreatic β-cells.

Incretin hormones and T2D.

GLP-1 may be able restore a Ca²⁺ signal by facilitating glucose-dependent K_ATP channel closure and by enhancing SOC allowing normal GSIS [64]. We were able to simulate this mechanism (Fig 6). In Fig 4 we suggested that in T2D glucose leads to an insufficiently increased ATP/ADP ratio (as a consequence of some unspecified dysfunction). We simulate the initial T2D conditions in Fig 6 (as well as in Fig 4) as decreased glucose dependent saturated ATP/ADP ratio that leads to a inability to increase [Ca²⁺]_c during GSIS. However, GLP-1 administration then leads to additional depolarization and [Ca²⁺]_c increase in our model (Fig 6). This additional depolarization is a consequence of increased I_SOC due to decreased [Ca²⁺]_ER after IP₃R activation by PKAa, that also leads to increased Ca²⁺ flux from ER to cytosol. PKA activation also decreases the conductance of K_ATP.

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Fig 6. Simulated response of GLP-1 action in T2D conditions.

T2D conditions were simulated by decreasing the glucose dependent saturated ATP/ADP ratio (ATD_m, Eq 4). ATD_m was decreased from 32 (basal level) to 15 as in Fig 4. In this case PM membrane potential is decreased and does not lead to [Ca²⁺]_c increase as glucose increases from 3 mM to 8 mM at 1000 sec as well as in Fig 4. GLP-1 administration was simulated at 3000 sec ([GLP-1] was increased from 3.1e-7 μM to 6.2e-4 μM. This induces cAMP increase with additional PM depolarization and Ca²⁺ influx into cell. Corresponding PKA activation also increases Ca²⁺ flow from ER. These effects lead to significant [Ca²⁺]_c and cAMP increase.

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Patients with T2D can have preserved GLP-1 and GIP secretion [142, 153]. However, the insulinotropic effect of incretins is diminished in T2D patients, due in part to reduced expression of incretin receptors as a consequence of perhaps glucotoxicity or lipotoxicity. For example, GLP-1R and GIPR levels were decreased in islets from mouse and rat model of diabetes and from T2D patients [57, 154, 155]. Elevated non-esterified fatty acids lead to a decreased GLP-1R expression and downregulation of GLP-1 receptor signaling in β-cell lines and mouse islets from db/db mice [156]. On other hand the transgenic expression of GLP-1R was able to restore GLP-1R-dependent stimulation of cAMP in isolated islets from mouse and in insulinoma cell lines [156, 157]. It was also shown that at high glucose levels GLP1-R can also be covalently modified in mouse islets by small ubiquitin-related modifier protein (SUMO) reducing its ability to be stimulated by GLP-1R agonist [56]. We have exploited our model to analyze these results. Proportional loss of cell surface GLP-1R and GIPR leads to a decreased cAMP and Ca²⁺ level even in the presence of high GLP-1 and GIP stimulation because the total receptor number is a limiting factor for activation of G-protein bound AC in our model. For example, a decrease in the total GLP-1R by 2-fold leads to a decrease of cAMP from 10.5 to 4.5 μM in a simulation of GLP-1 administration (if the calculations were made similar to Fig 3).

In summary, the results in this section suggest that the concentration of incretin hormone receptors on the PM can be a limiting factor in incretin function in rodent, normal human and T2D patients.

Catecholamines.

Catecholamines (epinephrine and norepinephrine), the agonists of alpha adrenergic receptors, inhibit insulin secretion by decreasing cAMP content in β-cell as a result of inhibition of AC activity [10]. Our simulation also shows a catecholamine inhibition of cAMP concentration that can reduce GSIS (Fig 7A). This is a consequence of block of PM bound AC (AC_P) activity because this AC isoform following catecholamine-receptor binding cannot produce cAMP (see Fig 7B). AC_P plays a role in maintaining low levels of cAMP at GSIS in our model even without GPCR agonists activating the cAMP pathway. This is a consequence of constitutive activity of GPCRs that activate AC_p. Even low cAMP levels may be necessary for insulin secretion (see above). Therefore, activation of catecholamine receptors decreasing AC_p activity further suppresses GSIS even though cAMP content may already be low.

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Fig 7. Simulation of catecholamines and GLP-1 interaction at high glucose levels.

(A) Cytoplasmic cAMP and [Ca²⁺]_c dynamics. (B) AC_GLP, AC_GIP and AC_AR are the concentrations of complexes of AC on PM (AC_P) bound with corresponding G-proteins activated by GLP-1R, GIPR and α_2A adrenergic receptors. (C) V_ACp is G-protein and Ca²⁺ dependent AC activity on PM. The initial simulation was with low glucose (3 mM) as in Fig 3. Increase of glucose (8 mM at 1000 sec) induced changes of intracellular parameters. Catecholamine concentration (AR₃) was increased from basal level (0.002 μM) to 3 μM at 3000 sec. Than GLP-1 was increased from basal level (3.1e-7 μM) to 6.2e-4 μM at 5000 sec.

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Subthreshold α₂-adrenergic activation with clonidine counteracts GLP-1 potentiation of GSIS [158]. Our simulation also shows that activated adrenergic receptors can block an increased cAMP concentration by incretin hormones because a significant part of PM bound AC is bound with G_i-proteins activated by catecholamine receptors. It blocks the ability of AC_P to be activated by G α_s-protein-coupled receptors. In this case the increase in cAMP concentration after simulation of GLP-1 was reduced (~6 μM, Fig 7A) in comparison with simulation in the absence of α₂-adrenergic activation (~15 μM, Fig 3C).

α_2A receptor antagonists have been proposed as T2D therapeutic agents [159]. We used our model to evaluate this possibility. Our simulation showed that a decrease in the number of α_2A receptors does not lead to a significantly increased cAMP concentration during GSIS, when we assume that α_2A receptor number is small (to reflect basal levels of this receptor accepted in our model). For example, with glucose stimulation (as in Fig 3) the increase in cAMP concentration is insignificant (from 2.104 to 2.106 μM) if the α_2A receptor content (Re_ARt in Eq 48) is lowered 10-fold. This happens because constitutive catecholamine activity blocks only a small fraction of the PM bound AC in the absence of specific activators (see Fig 5B). Block of this fraction cannot significantly effect activation of the remaining fraction of AC_P by incretins (not shown). Consequently, α_2A receptor antagonists are unlikely to be useful in T2D if the activity of α_2A receptors is insignificant.

However, T2D diabetes risk can be associated with increased expression of α_2A receptors and a concomitant reduction in insulin secretion [160]. Our simulation of cAMP dynamics with an increased α_2A receptor expression level (even without their agonists being present) gives a result that is indeed similar to α₂-adrenergic activation shown in Fig 7, i.e. to significant decrease in cAMP. This shows that increased α_2A receptor expression should lead to decreased cAMP production during GSIS without and with the incretin hormones (under physiological conditions). AC_P activity is reduced since a significant part of PM bound AC may be inhibited (bound with G_i-proteins) and cannot be activated during GSIS or by incretin receptors. In this case α_2A receptor antagonists might be useful in increase insulin secretion. It does seem that this effect is limited to situations where this is significantly increased expression of α_2A receptors over control situations.

Regulation of AC and PDE activity.

Regulation of intracellular cAMP concentration during GSIS is also possible using AC activators such as forskolin or a PDE inhibitor (e.g., IBMX) [139]. For example, β-cell PDE3B can be activated by insulin, IGF-1 and leptin that inhibit insulin secretion [140]. Interestingly, PDE4 inhibitors may increase insulin secretion in humans [161].

Increased cAMP by AC activation or PDE inhibition causes various events, including Ca²⁺ mobilization from internal Ca²⁺ stores and activation of non-selective cation channels [80, 84]. In our model, activation of AC or PDE inhibition increased cAMP concentration, similar to the above for GLP-1 or GIP stimulation (see below for a special case of expression of different PDE isoforms). Then increased cAMP leads to PKA activation and block of K_ATP channels, activation of Ca²⁺ release from ER and additional PM depolarization (see Figs 1 and 6). Notice that dual activation of the PKA and Epac branches of the cAMP signaling mechanism can act directly on the exocytotic machinery [68, 146].

FFAR1/GPR40 receptops.

FFAR1/GPR40 agonists enhance insulin secretion. They markedly enhance cytoplasmic and mitochondrial Ca²⁺ in insulinoma cells and increased IP₃ and ATP levels in islets normal and diabetic rats [164]. GSIS (i.e. at high glucose level) was also augmented in pancreatic β-cells in normal and diabetic mice that overexpressed FFAR1/GPR40 [165, 166].

We simulated the effect of FFAR1/GPR40 activation via Gα_q-proteins leading to PLC (PLC_p) pathway activation (Fig 8). However, this effect was insignificant for Ca²⁺ and IP₃ at low glucose levels and only increased glucose levels (that can increase [Ca²⁺]_c) can significantly activate PLC_p (Fig 8D). This leads to an additional rise in IP₃ and DAG concentrations. Increased IP₃ also results in decreased [Ca²⁺]_ER and an increase in [Ca²⁺]_c as a result of IP₃-dependent release of Ca²⁺ from the ER. Decreased [Ca²⁺]_ER also opens store-operated channels (SOC) accelerating Ca²⁺ and possibly other cations into the cytoplasm through these channels. PLC_p activation decreases PIP₂ that can also lead to additional K_ATP channels inhibition (see Fig 8E). All these processes enhance PM depolarization, increasing Ca²⁺ influx (Fig 8A). Formation of DAG also activates PKC, resulting in increased efficiency of Ca²⁺ on exocytosis during GSIS (Fig 1).

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Fig 8. Effects of FFAR1/GPR40 activators and acetylcholine (Ach) on changes of intracellular parameters.

Two protocols were stimulated: 1. (---) glucose was increase from 3 mM to 8 mM at 3000 sec, than acetylcholine, M₃R activator, (AM₃ concentration in model) was added at 5000 sec (AM₃ was increased from basal level 0.0022 μM to 2.2 μM) and 2. (- - - -) FFAR1/GPR40 agonist was added at 1000 sec (AR₇ was increased from basal level 0.0506 μM to 82 μM), than glucose (3000 sec) and acetylcholine were added (5000 sec). (A) Free cytoplasmic Ca²⁺ concentration ([Ca²⁺]_c. (B) Free Ca²⁺ concentration in ER ([Ca²⁺]_ER); (C) IP₃ concentration. (D) V_PLP is the activity of PM bound PLC (Eq 78). (E) Concentrations of PIP₂ on PM.

https://doi.org/10.1371/journal.pone.0152869.g008

These simulations reflect experimental data. Unbound FFAs in plasma in the physiological range can activate FFAR1/GPR40 in the presence of glucose and eventually amplifies GSIS in short-term exposure [106, 167]. FFA action on FFAR1/GPR40 caused PLC activation and a rise in [Ca²⁺]_c both in primary mouse and rat β-cells and in INS-1 cells with increased glucose [168–170]. Palmitate also markedly reduced the calcium storage capacity of the ER measured by thapsigargin-induced Ca²⁺ release, suggesting that ER calcium stores may participate in raising [Ca²⁺]_c in isolated mouse islets [37]. These events are consistent with a decrease in [Ca²⁺]_ER following FFAR1/GPR40 activation in our simulation (Fig 8B).

We evaluated the effect of FFAR1/GPR40 potentiation at low glucose in more detail to evaluate possible enhanced risks of hyperglycemia. At low glucose concentrations, activating FFAR1/GPR40 does not increase [Ca²⁺]_c in pancreatic rat β-cells [170] nor does it stimulate insulin secretion in mouse islets [166]. Interestingly, mouse pancreatic β-cells expressing increased FFAR1/GPR40 do not show increased insulin secretion in low glucose [165]. PLC activity is unaffected in our model (Fig 8D left part), because increased cytosolic Ca²⁺ is required to increase G-protein dependent PLC activity. This is an advantageous feature that should reduce the chance of producing hyperinsulinemia and hypoglycemia in T2D patients given FFAR1/GPR40 agonists. Indeed, selective pharmacological activation of FFAR1/GPR40 by TAK-875 significantly improved glycemic control in T2D patients with only a modest risk of hypoglycemia [171]. This result resembles our analysis of GSIS stimulated by GLP-1, where GLP-1 action alone (i.e. at low glucose) has little or no effect on insulin secretion. Only a combination of GLP-1 and high glucose potentiates insulin secretion. We explained this effect also through the dependence of increased [Ca²⁺]_c to increase G-protein dependent AC activity.

We can also predict that using FFAR1/GPR40 agonists simultaneously with K_ATP channel blockers can lead to increased incidence of hypoglycemia similar to employing GLP-1R agonists and K_ATP channel blockers together. Both of these effects are associated with an increase in [Ca²⁺]_c due to sulfonylureas or similar K_ATP inhbiting drugs that can activate PLC or AC even in low glucose levels. Indeed, FFAR1/GPR40 activation increased insulin secretion in the presence of sulfonylurea under low glucose conditions through enhancement of PKC signaling in insulinoma cells [172].

Activation of FFAR1/GPR40 may be a viable therapeutic approach for treatment of type 2 diabetes [173]. Our simulations suggest that FFAR1/GPR40 should increase insulin secretion in the presence of elevated glucose, because FFAR1/GPR40 agonists should increase [Ca²⁺]_c and DAG during GSIS, supporting the evidence that these agonists may be useful as novel insulin secretagogues with low risk of hypoglycemia. However, the simulations also indicate that using FFAR1/GPR40 activators together with sulfonylureas could increase the incidence of insulin secretion overshoot and thereby increase hypoglycemia, as it was discussed for incretin hormones.

Activation of muscarinic receptors.

Cholinergic muscarinic agonists, including the endogenous neurotransmitter acetylcholine and its synthetic nonhydrolyzable analog carbachol, increase [Ca²⁺]_c and GSIS in normal β-cells [3]. Stimulation of the MR activates Gα_q-proteins leading to PLC_p activation in our model (Fig 1) Modeling of MR activation is shown in Fig 8 (right) for high glucose levels for two cases, with and without preliminary FFAR1/GPR40 activation. This simulation (without preliminary FFAR1/GPR40 activation) corresponds to experimental evidence that at high glucose level, stimulation with the muscarinic receptor agonists (for example, carbachol), induces a rapid and sustained PLC activation [124, 174], depolarizes PM and enhances glucose-induced electrical activity [175]. Several studies have also suggested that acetylcholine can regulate [Ca²⁺]_c dynamics leading to [Ca²⁺]_c increase and [Ca²⁺]_ER decrease [129, 176]. Our model also takes into account a suggestion that the depolarizing effects of the muscarinic receptor agonists can be attributed partially to a store-operated current (SOC) that activates following IP₃-dependent release of Ca²⁺ from the ER.

β-cell specific inactivation of genes encoding Gα_q/11 in mice leads to decreased PM depolarization with increased glucose [108]. According to our simulations this result seems to be a consequence of decreased SOC because (1) [Ca²⁺]_ER does not decrease significantly without IP₃ and (2) K_ATP channels remain open because PIP₂ concentration does not decrease markedly without PLC activation.

The effect of glucose and carbachol on GSIS is at least additive in rat islets [115]. Based on our model this effect can be explained by significant independence of PLC pathway that increases IP₃ and DAG (DAG content changes proportionally IP₃ in our model) (Fig 8C) and glucose activation of GSIS through PM depolarization and [Ca²⁺]_c increase (Fig 8A).

Moderate levels of acetylcholine have little effect on PM depolarization or on sustained cytosolic Ca²⁺ in mouse islets at low glucose [124, 176–178]. We simulated activation of muscarinic receptors (MR) at low glucose levels. In this case, activation of MR evoked only a small increase in [Ca²⁺]_c similar to modeling of FFAR1/GPR40 activation (for example, see Fig 8, left) if conductivity of specific nonselective cation channels (NALCN) (that can be activated by MR) was insignificant (not shown). This can be explained as well as for FFAR1/GPR40 activation (see Fig 8) so that increased cytosolic Ca²⁺ is necessary for activation of membrane bound PLC.

A different scenario was found for rat or human islets. Acetylcholine increased [Ca²⁺]_c in isolated islets from Wistar rats at low glucose (2.8 mM) [179] and in human β–cells [62]. These results can be explained by activation of the MR dependent NALCN channels that lead to transient PM depolarization and increased [Ca²⁺]_c even with the simulation conducted in low glucose (Fig 9). Likewise, our simulation of electrophysiological events in human β-cells showed that activation of NALCN channels can lead to additional PM depolarization and increased insulin secretion at high glucose. This can also decrease the threshold for glucose-induced spike activity and also lead to increased [Ca²⁺]_c [30].

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Fig 9. Effects of Ach on changes of intracellular parameters at activated NALCN channels at low glucose.

(A) PM potential (V_p); (B) [Ca²⁺]_c and [Ca²⁺]_ER; (C) Cytoplasmic IP₃ concentration and the activity of PM bound PLC (V_PLP). For simulation of NALCN channel activity the maximal conductance (g_mNM3, Eq 25, CM) was initially increased from 0 nS (basal level), to 35 nS. Addition of acetylcholine that opens NALCN channel was simulated at 1000 sec at the arrow (AM₃ was increased from basal level 0.0022 μM to 2.2 μM). All simulations were performed at low glucose (3 mM).

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Acetylcholine activates Na⁺ influx by an activation of NALCN permeable to Na⁺ [3], i.e. an increased Na⁺ inward current through NALCN leading to PM depolarization (see [30, 50]) that corresponds to mechanisms considered in our model, where opening of NALCN lead to increase in Na⁺ influx (see Eq 25, CM).

Activation of MR dependent NALCN channels in low glucose could be a mechanism by which acetylcholine increases human insulin secretion during first phase of feeding where an increase in insulin secretion precedes an increase in blood glucose. This increase, termed "cephalic phase", is thought to be largely mediated by the release of acetylcholine from nerves innervating pancreatic islets or from the alpha cells [6, 62, 180]. In this case activation of MRs may increase the capability for insulin secretion even while still in sub-threshold glucose levels.

Zawalich and colleagues [115] showed that the failure of PLC activation resulted in impairment of insulin secretion under sustained glucose exposure in rats. It was also found that mice that selectively lacked MRs in pancreatic β-cells showed decreased insulin release and impaired glucose tolerance. Consistent with this concept, transgenic mice that selectively overexpressed type 3 MRs in pancreatic β-cells exhibited increased insulin release and improved glucose tolerance [181]. Our simulation with a decreased number of MRs leads to decreased [Ca²⁺]_c and DAG concentration at high glucose because constitutive MR activity takes part in IP₃ and DAG production. For example, our simulation with 10x reduced MR abundance (to compare with the example in Fig 3) shows a significant decrease in [Ca²⁺]_c (from 272 to 250 nM) and DAG concentration (from 1131 to 926 #/μm^-2) at high glucose. The "cephalic phase" should also be blocked in these conditions. This should lead to decreased insulin secretion. These experimental data and our simulations suggest that optimal acetylcholine receptor stimulation is essential for β-cell insulin secretion.

Approaches aimed at enhancing signaling through β-cell MRs (or downstream) might become therapeutically useful for the treatment of T2D (see [104, 182]). However, our consideration shows that MR activation could conceivably lead to hypoglycemia at relatively low glucose concentrations as a consequence of NALCN channel activation leading to PM depolarization and increased [Ca²⁺]_c.

FFAR1/GPR40 and muscarinic receptor interactions.

The temporal interplay between FFAR1/GPR40 and muscarinic receptors was investigated [37]. Palmitic acid at physiological levels could acutely inhibit subsequent acetylcholine-stimulated insulin secretion (through activation of FFAR1/GPR40) and the rise of intracellular Ca²⁺ production in isolated mouse islets during GSIS. According to these authors the mechanism of such effect could be a Ca²⁺ release from the ER during FFAR1/GPR40 activation that prevents subsequent acetylcholine induced Ca²⁺ release. Our simulation shows that this effect can take place, i.e. FFAR1/GPR40 activation can prevent additional [Ca²⁺]_ER decrease and Ca²⁺ release at acetylcholine administration (Fig 8A and 8B). However, there is also competition at the level of PLC such that activation of FFAR1/GPR40 leads to G-proteins binding to PLC_p that decreases the ability of PLC_p to bind with G-proteins, activated by acetylcholine (Fig 8D). Consequently, simultaneous use of activators for FFAR1/GPR40 and the muscarinic receptor may not lead to significant additive effects on insulin secretion.

Plasma membrane potential.

Increased extracellular glucose promotes membrane depolarization. Several ionic channels regulate this process. A diagram of the main β-cell specific channels considered here is presented in Fig 1. We used our models of electrophysiological processes in the β-cell [29, 30, 33] as a framework. However, only the most important currents were taken into account to achieve the desired granularity. For simplification Boltzman-type equations in steady-state were employed for activation (d_i) and inactivation functions (f_i) to avoid a simulation of spike activity.

We modeled the electrophysiological events for one cell. The plasma membrane (PM) potential in a single β-cell can be described with the following current balance differential equation (8) where V_p is the PM potential, t is the time, C_m is the whole-cell membrane capacitance, I_KATP is the ATP-sensitive K⁺ channel current, I_Kr is the voltage-dependent K⁺ current, I_Ca is the high-voltage-activated Ca²⁺ current, I_SOC is the store-operated current activated by decrease of Ca²⁺ in ER, I_PCa is the plasma membrane Ca²⁺-pump current, I_Nab is the Na⁺ background current, that can be activated also by muscarinic acetylcholine M₃ receptor.

ATP-sensitive K⁺ channels current (I_KATP).

We adopted a kinetic model [29, 33] for whole-cell ATP-sensitive K⁺ channels current (I_KATP). However, a dependence of this current on PIP₂ and PKA activation was also found and we introduced the corresponding factors. The influence of PIP₂ was represented as Michaelis-Menten function (f_KPI, Eq 10) where PIP₂ increase can enchance I_KATP.

Both SUR1 and K_ir6.2 subunits contain consensus PKA phophorylation sites and, for example, GLP-1 dependent activation of PKA leads to phosphorylation of SUR1 subunits lowering their affinity for ADP [205, 206]. For simplicity we introduced this dependence as a function (f_KPKa, Eq 13) where an activated PKA (PKA_a) decreases the calculated [MgADP_f]_c leading to additional I_KATP decrease. (9) where (10) (11) (12) (13) (14) where g_mKATP is is the maximum conductance for I_KATP, g_mKATPi is the initial conductance, g_mKATP1 is the increase in conductance in time (t9). E_K is the reversal potential for K⁺ current, PIP₂ is the PIP₂ content on PM, K_KPI2 is the activation constant, O_KATP is the fraction of open K_ATP channels, k_dd, k_td and k_tt are the dissotiation coefficients, MgADP_f is free Mg-bound ADP, ATP_f and ADP_f are free ATP and ADP concentrations in cytoplasm; k_KPKa is the inhibition constant.

To determine K_KPI2 (Eq 10) we suggest that f_KPI dependence corresponds to PIP₂ dose-dependent activation of K_ATP channels (in absence GST-Syn-1A) where EC₅₀ values for PIP2 was calculated as 2.38±0.81 μM [38] or 1125 #/μm^-2 in our definition (see Eq 1).

Voltage-gated K⁺ current (I_Kr).

Different voltage gated K⁺ currents (I_Kr) were also registrated in β-cell. For example, delayed rectifier K⁺ current is main voltage-dependent K⁺ current in rodents [207]. This current activates at membrane potential near -30 mV and then increased with the applied voltage, an inactivation was negligible during at least 200 ms depolarization [208]. The Hodgkin-Huxley-type with stationary dependences for delayed rectifier K⁺ current and some corresponding coefficients were used from our previous mouse and human β-cell model to describe I_Kr [29, 30]. (15) where (16) (17) g_mKr is the maximum conductance for I_Kr, V_dKr is the half-activation potential, k_dKr is the slope of half-activation potential.

Ca²⁺ current (I_Ca).

I_Ca was modeled as L-type Ca²⁺ channel with one inactivation gating variables [29] without time dependent parameters (18) where (19) (20) where g_mCa is the maximum conductance for I_Ca, E_Ca is the reversal potential for Ca²⁺ current, V_dCa is the half-activation potential, k_dCa is the slope of half-activation potential, k_dCap is the coefficient of the constitutive channel activity, V_fCa is the half-inactivation potential, k_fCa is the slope of half-inactivation potential.

Store-operated Ca²⁺ current (I_SOC).

[Ca²⁺]_ER decrease activates the Ca²⁺ release-activated Ca²⁺ current through PM. Interestingly, in human Jurkat leukaemic T cells expressing an ER-targeted Ca²⁺ indicator, SOC channel activation follows the function of [Ca²⁺]_ER, reaching half-maximum at ~200 μM with a Hill coefficient of ~4 [196]. We use these dependence and coefficients in our model (Table 2). We consider I_SOC as a voltage-independent Ca²⁺ inward current (21) where (22) g_mSOC is the maximum whole-cell conductance, f_SOC is the [Ca²⁺]_ER dependent function, K_NS is the [Ca²⁺]_ER inhibition constant.

Plasma membrane Ca²⁺ pump current (I_PCa).

Ca²⁺ pumps provide an outward current and also contribute to V_p. The corresponding equation was adapted from previous model [33] (23) where P_mCa is the maximum I_PCa current, K_pCa is the [Ca²⁺]_c activation constant.

Na⁺ background current (I_Nab).

The model contains a voltage-independent Na⁺ background current. This current can depolarize the resting PM potential and modifies mouse and human β-cell electrical activity (see for details [29, 30, 33]). We have also included the specific NALCN channels, that activatable by the muscarinic acetylcholine M₃ receptor. Than (24) where (25) E_Na is the reversal potential for Na⁺, g_mNb is the permanent conductance for I_Nab, g_mNM3 is the maximum conductivity of NALCN channels and LRG₆ is the M₃ receptor bound with ligand and G-protein (see below).

Receptor dynamics and G protein cascade.

Our model of receptor dynamics is based on several earlier studies [209, 210]. However, we were not able to find a model that matches well to our aims. The majority of these models do not include a G-protein bound with target enzyme. For this reason we constructed a minimal model of the receptor signaling transduction and corresponding target enzymes activations.

Several suggestions were made. Our model assumes that initial concentration of ligand is sufficiently high so that it is not significantly depleted by binding to the receptors and therefore the ligands can be presumed as a fixed concentration. The process of receptors and G-proteins synthesis is assumed to compensate for intracellular receptor and G protein degradation, so that the total area density of receptors and G proteins in different states (free, active, and internalized) remains constant.

We have assumed that the processes occur in the plasma membrane and all the species, except for the agonist, are membrane bound and are diffuse freely. We simplify the model by ignoring the processes, such as the nucleotide exchange on G-proteins that are not bound to the receptor or bound to the receptor that is not bound to the ligand and others. Receptor desensitization was included as a process that can deactivate the receptor-ligand complex. The multiple deactivation and reactivation steps were shortened in single irreversible reactions (see Fig 13).

We modeled two different models of ligand-receptor interactions: 1. Collision coupling model and 2. Pre-coupling model (see Fig 13). Based on the above assumptions and using the law of mass action the collision coupling model suggests that ligands, receptors and G-proteins collide and binds to the corresponding enzyme transiently to produce target enzyme activation. For simplicity, we use concentrations on PM without brackets. Than a collision coupling model is represented as a system of ordinary differential equations: (26) (27) (28) The following conservation constrain also holds (R_nt) (29) or (30) where [L_n] is the free ligand concentration; K_L is the Michaelis-Menten constant, R_n, LR_n, LRG_n and R_nd are, respectively, the area density of free, bound with ligand, bound with ligand and G-protein and internalized (desensitized) receptors and R_nt is the total receptor area density in # μm^-2. (Definition L_nR_nG_n instead L_nR_nG_nGDP was used for simplification). k_1n, k_2n, k_3n, k_8n and k_9n are the forward rate constants and k_2nr is the backward rate constant (see Fig 13A).

The first binding step reflects a simple bimolecular interaction between the ligand (L_n) and the receptor (R_n) (coefficient k_1n) that is strictly dependent on ligand concentration. According to mass action kinetics, we have assumed that G_αnGTPE_n are produced at a rate proportional to the area density of GTP-bound α subunit of G-protein, G_αnGTP, and the inactive target protein area density (E_fn) (coefficient k_4n). (31) (32) (33) where k_4n - k_7n are the forward rate constants.

We suggest that the content of the endogenous G-proteins (G_nt) and target enzymes (E_nt) remain constant (34) or (35) (36) (37) or (38) where G_nGDP, LRG_n, G_αnGTP, G_αnGDP and G_αnGTPE_n are the area density of free, bound with ligand and receptor, active, inactive and bound with the target enzyme G-proteins, respectively. G_nt is the total area density of G_n proteins per unit of PM, both activated and inactivated. E_fn is the free target enzyme and E_nt is the total area density of the target enzyme.

Pre-coupling mechanism of ligand-receptor interaction suggestes that the stable receptor/G-protein complex exists in the absence of ligand and ligand bounds with this complex (see Fig 13B). Equations for a pre-coupling model can also be obtained similarly Eqs 26–30 as a system of ordinary differential equations: (39) (40) (41) The following conservation constrains also hold (42) (43) (44) or (45)

Other equations for pre-coupling model are similar to the collision coupling model (Eqs 31–38).

Several suggestions were accepted in order to determine of coefficients of both models:

The number of βγ subunits is the same as the number of α subunits, because each G-protein splits into one α and one βγ. Estimated ratio of G-protein to receptors was accepted as ~ 10:1 [211, 212].

Kinetics of ligand binding are usually fast relative to the other processes in the model (see [213]). The fast speed observed for receptor activation in response to saturated concentrations of small agonist molecules (coefficient k_1n, Fig 13A) has half-life (t_1/2) ranges between ≈ 40 and 100 ms in the case of α_2A- and β₁-adrenergic receptors, muscarinic M₁- and M₂-receptors [214] (k = ln2/t_1/2). We used a coefficient of receptor ligand binding (k_1n ≈ 7 μM^-1 s^-1) for the collision coupled model that corresponds to the half-life about 100 ms.

Coefficient for binding of LR_n with target G protein in the collision coupling model (k_2n in Fig 13A) depends on the area density of moleculs on PM and was evaluated as 1 #^-1μm² s^-1 [215]. Coefficient for receptor–G-protein unbinding (k_2nr) was calculated for collision coupling model as 0.68 s^-1 and for pre-coupling model as 6.8 s^-1 [216].

Speed of G-protein activation (coefficient k_3n Fig 13A and 13B) is slow (half-life t_1/2≈ 1 sec) [214]. This values corresponds k_3n ≈ 0.7 s^-1. It is close to 0.8 s^-1 evaluated by [217] or 1 s^-1 [218].

Binding of G_αnGTP with target enzyme E_fn (coefficient k_4n in Fig 13A and 13B) was evaluated by [216] as 1 #^-1μm² s^-1. Deactivation of G_αnGTP through its own GTPase activity (coefficient k_5n Fig 13A and 13B) was evaluated by [216] as 0.026 s^-1. Dissociation of complex of G protein and target enzyme (coefficient k_6n Fig 13A and 13B) was evaluated by us as 0.4 s^-1. The recombination of α and βγ subunits (coefficient k_7n in Fig 13A) is thought not to be limiting and was evaluated as 1 μm² s^-1 [216, 218]. Typical rate constants for desensitization of activated receptor (k_9n) and its return to the surface (k_8n) in our model was taken from model for slow GLP-1 receptor kinetics [24] (k_8n = 5 10⁻⁵ s^-1 and k_9n = 0.000283 s^-1). We used these parameters and coefficients if no additional experimental data can be found in literature for some receptors, corresponding G proteins or target enzymes.

Tonic receptor effects are simulated by including the presence of low concentrations of corresponding receptor ligands even when the experiments were simulated without specific agonists.

GLP-1R (R₁).

We used the collision coupling model for this receptor. The processes were described similar to Eqs 26–40) (46) GLP1 was used instead [L_n], GLP1i is the initial GLP-1 concentration. GLP11 is the increase in GLP1 in time (t2), K_GLP1 was used instead K_L and R_1X instead the corresponding coefficients (see Table 3). AC_GLP is the PM bound AC activated by GLP-1 receptor. The parameters and coefficients can be identified by their subscripts, where 1 was used instead n (Table 3).

Specific concentrations and coefficients: According [219] the amount of GLP-1R (Re_GLt) in one INS-1 cell is about 2,000, i.e. it can be evaluated as about 2 # μm^-2 (if cell surface is about 1000 μm^-2). Half saturation (log₁₀M = -10.5) for GLP1 was measured as an increase in cAMP production in the pancreatic β-cell line (Fig 5 from [220]), that we used for evaluation of K_GLP1 (0.000031 μM, Table 3). Other coefficients were chosen as general coefficients.

GIPR (R₂).

Regulated mechanisms for GIP receptor are assumed to be similar to those of GLP-1R. (47) GIP was used instead [L_n], K_GIP instead K_L and R_2X instead the corresponding coefficients (see Table 3). AC_GIP is the PM bound AC_p activated by GIP receptor. GIP_i is the initial GIP concentration. GIP1 is the increase in GIP in time (t3). The parameters and coefficients can be identified by their subscripts, where 2 was used instead n (Table 3).

Specific concentrations and coefficients: INS-1 cells expressed an average of 2443μ400 GIP receptors (G_2t) on the cell surface or ~ 2.5 # μm^-2 [221]. EC₅₀ for GIP was evaluated as 171 pM for activation cAMP production [222], i.e. K_GIP = 0.000171 μM. Other coefficients were chosen as the general coefficients.

α2A-adrenergic receptor (R3).

A model of α_2A adrenergic receptor activation in β-cell was constructed as a collision coupling model. In this case an active G_αGTP messenger (G_αT3) released from stimulated receptor binds to the enzyme AC_pf. (48) AR₃ was used instead [L_n], K_AR3 instead K_L and R_3X instead the corresponding coefficients (see Table 3). AR_3i is the initial AR₃ concentration. AR₃₁ is the increase in AR₃ in time (t4), AC_AR is the PM bound AC activated by α_2A-adrenergic receptor. The parameters and coefficients can be identified by their subscripts, where 3 was used instead n (Table 3).

Specific concentrations and coefficients: Noradrenaline (norepinephrine) was shown to be a potent inhibitor of GSIS from rat pancreatic islets, with half-maximal inhibition of the secretory response to 20 mM-glucose occurring at approx. 0.3 μM [223], and we accepted this value for the coefficient K_AR3. Other coefficients were chosen as the general coefficients.

Basal concentrations of GLP-1R, GIPR and AdR, that determinate constitutive receptor activity, were taken in such manner that 0.1 AC_P was bound for each receptor (see Eq 59).

M3 muscarinic receptor (R₆).

We used the pre-coupled model for this receptor. (49) AM₃ was used instead [L_n], K_AM3 instead K_L and R_6X instead the corresponding coefficients (see Table 4). PLC_M3 is the PM bound PLC activated by M₃ muscarinic receptor; PLC_pf is the free PLC on PM (see Eq 88). AM_3i is the initial AM₃ concentration. AM₃₁ is the increase in AM₃ in time (t6). The parameters and coefficients can be identified by their subscripts, where 6 or M₃ were used instead n.

Specific concentrations and coefficients: acetylcholine and carbachol (MR agonist) have different K_AM3. We used the coefficients by Hoffman et al [224] for acetylcholine in human M₃ muscarinic receptor (see Table 4). Agonists such as acetylcholine, carbachol or muscarine activate each receptor construct with half-maximal activation times between 60 and 70ms for human M3 muscarinic receptor [225] (coefficient R₆₁) (or R₆₁ = 10 s^-1).

Receptor deactivation kinetics was found to be slow and independent of agonist concentrations [224] (coefficient R_62r was taken according [216]). R₆₈ (coef k_8n in Fig 13B) was evaluated as the coefficient of M₃ receptor dephosphorylation in work [226]. Human M₃ receptors were internalized with a half life of 11 min (660 s) for M₃ receptors expressed in COS-7 cells and in the presence of 10⁻³ M carbamylcholine (for receptor activation) [227] and we determined the coefficient R₆₉ from these data. Other coefficients were chosen as general coefficients (Table 4).

FFAR1/GPR40 (R₇).

A model for FFAR1/GPR40 was made as for collision coupling model receptors. (50) AR₇ was used instead [L_n], K_AR7 instead K_L and R_7X instead the corresponding coefficients (see Table 4). PLC_R40 is the PM bound PLC activated by FFAR1/GPR40 receptor. AR_7i is the initial AR₇ concentration. AR₇₁ is the increase in AR₇ in time (t7). The parameters and coefficients can be identified by their subscripts, where 7 or R40 were used instead of n.

Specific concentrations and coefficients: K_AR7 for AR7 was evaluated as palmitic acid potency (EC₅₀) to induce [Ca²⁺]_c rise in mouse CHO cells expressing FFAR1/GPR40 [167]. Other coefficients were chosen as the general coefficients.

Basal concentrations of FFAR1/GPR40 and AR₇, that determinate constitutive receptor activity, were taken in such manner that 0.1 PLC_Pt was bound for each receptor (see Eq 79).

cAMP dependent PKA and Epac activation.

Binding of cAMP to the regulatory units of PKA or Epac results in release of the catalytic units and an activation of PKA or Epac. We used the mathematical models of PKA activation [198]. Relative steady-state level of PKA activation by cAMP (f_pca) was described by an empirical Hill-type equation [25]. (67) where (68) (69) where PKA_a is the relative concentration of active PKA, PKA_i is the relative concentration of inactive PKA, the total relative concentration of active and inactive PKA was accepted as 1, t_pka is the time constant; k_ak is the scaling factor, f_pca is cAMP potential factor, where K_pcm is the cAMP activation coefficient [230] and hpca is the Hill coefficient.

The pathway, that includes Epac activation, was modeled similarly to the PKA dynamics (70) where (71) (72) where EP_a is the relative concentration of active Epac, EP_i is the relative concentration of inactive Epac, the total relative concentration of active and inactive Epac was accepted as 1; f_EcA is the cAMP potential factor, where K_mep is the cAMP activation constant for Epac, hce is the Hill coefficient, t_ep is the time constant.

Epac has a lower binding capacity for cAMP to compare with PKA [94] (see Table 5).

PIP₂ dynamics.

According to [183] ATP dose-dependently stimulated PIP₂ synthesis has the half-maximally stimulation at 300 μM ATP that significantly lower than ATP concentration in cytoplasm (see [31]). For this reason we did not take into account ATP concentration dependence at PIP₂ synthesis. On the other hand PLC hydrolyzes plasma PIP₂ molecules into inositol trisphosphate (IP₃) and DAG. The rate of this reaction depends on activity of PLC. (75) where (76) where V_PL is the PLC activity (see below, Eq 77), K_PIP2 is the PIP₂ activation coefficient.

Density of phosphoinositides at the plasma membrane in β-cell remains uncertain. We assumed that free PIP₂ is about 5000 # μM^-2 in our modeling similarly evaluation for tsA201 cells in work [201].

IP₃ dynamics.

The generation of IP₃ is determined by the hydrolysis rate of PIP₂. Because IP₃ molecules diffuse through the cytosol, we refer to IP₃ production and degradation in terms of concentration (in μM) instead of the area density of molecules on PM. Different mechanisms seem to be available in the β-cells for degradation of IP₃, however, we assume that IP₃ is degraded at a rate proportional to the concentration of IP₃. Then we used the simplest model of IP₃ dynamics from [33]) but it was modified to consider the processes of IP₃ production on PM. (82) where f_Nc is the coefficient to convert from # μm^-2 to μM (Eq 1), k_dIP3 is the rate constant of IP₃ degradation.

DAG dynamics.

DAG is the product of PLC-catalyzed breakdown of phosphoinositides, stimulates by PLC where it is produced stoichiometrically with IP₃. However, DAG is bound with PM. Model of the DAG dynamics was developed similarly to the IP3 model. Coefficients from model [201] for native tsA201 cells were used (Table 6). (83) where k_dDAG is the coefficient of DAG degradation

DAG dependent PKC activation.

Binding of DAG to the regulatory units of PKC results in release of the catalytic units from PKC and its activation. Relative PKC activation by DAG was described similarly to work [202] as equation: (84) (85) where PKC_t = 1

PKC_a is the relative concentration of active PKC, PKC_i is the relative concentration of inactive PKC, PKC_t is the total relative concentration of active and inactive PKC, k_PKC and k_PKCr are the forward backward rate constants. t_1/2 was evaluated as 204 s^-1 [202] (k_PKCr = 0.0034 s^-1).

IP₃ Receptor.

The simplest model of IP₃ receptor (IP₃R) activated by IP₃ and Ca²⁺ was taken from Bertram and Sherman [203]. Additional possibility is an activation of IP₃R by PKA. PKA-mediated phosphorylation leads to a direct increase in the sensitivity of the IP₃ receptor toward IP₃ without shifting its Ca²⁺ sensitivity. We modeled this effect as a decreased IP₃R activation constant with PKA activation (function f_IPKA). (90) where

P_RIP3 is the channel open probability, K_RPCa is the [Ca²⁺]_c activation constant, K_IP3 is the IP₃ activation constant, d_inac is the [Ca²⁺]_c inhibition constant at higher concentrations of Ca²⁺, k_IP3R is the PKAa activation coefficient. The first factor represents an activation by Ca²⁺, the second an activation by IP₃, and the third an inactivation at high concentration of [Ca²⁺]_c.