Mathematical modelling of local calcium and regulated exocytosis during inhibition and stimulation of glucagon secretion from pancreatic alpha‐cells

Abstract

Key points

• The control of glucagon secretion from pancreatic alpha‐cells is still unclear and, when defective, is involved in the development of diabetes.

• We propose a mathematical model of Ca²⁺ dynamics and exocytosis to understand better the intracellular mechanisms downstream of electrical activity that control glucagon secretion.

• The model exploits compartmental modelling of Ca²⁺ levels near open and closed high voltage‐activated Ca²⁺ channels involved in exocytosis, in the sub‐membrane Ca²⁺ compartment, in the bulk cytosol and in the endoplasmic reticulum.

• The model reproduces the effects of glucose, glucagon‐like peptide 1 (GLP‐1) and adrenaline, providing insight into the relative contributions of the various subcellular Ca²⁺ compartments in the control of glucagon secretion.

• Our results highlight that the number of open Ca²⁺ channels is a dominant factor in glucagon release, and clarify why cytosolic Ca²⁺ is a poor read‐out of alpha‐cell secretion.

Abstract

Glucagon secretion from pancreatic alpha‐cells is dysregulated in diabetes. Despite decades of investigations of the control of glucagon release by glucose and hormones, the underlying mechanisms are still debated. Recently, mathematical models have been applied to investigate the modification of electrical activity in alpha‐cells as a result of glucose application. However, recent studies have shown that paracrine effects such as inhibition of glucagon secretion by glucagon‐like peptide 1 (GLP‐1) or stimulation of release by adrenaline involve cAMP‐mediated effects downstream of electrical activity. In particular, depending of the intracellular cAMP concentration, specific types of Ca²⁺ channels are inhibited or activated, which interacts with mobilization of secretory granules. To investigate these aspects of alpha‐cell function theoretically, we carefully developed a mathematical model of Ca²⁺ levels near open or closed Ca²⁺ channels of various types, which was linked to a description of Ca²⁺ below the plasma membrane, in the bulk cytosol and in the endoplasmic reticulum. We investigated how the various subcellular Ca²⁺ compartments contribute to control of glucagon‐exocytosis in response to glucose, GLP‐1 or adrenaline. Our studies refine previous modelling studies of alpha‐cell function, and provide deeper insight into the control of glucagon secretion.

Abbreviations

CFTR

cystic fibrosis transmembrane conductance regulator

Epac2

exchange protein directly activated by cAMP isoform 2

ER

endoplasmic reticulum

FFA

free fatty acid

GLP‐1

glucagon‐like peptide 1

GS

glucagon secretion

GSR

glucagon secretion rate

HVA

high voltage‐activated

KATP channels

ATP‐sensitive K⁺channels

PKA

protein kinase A

SOC

store‐operated Ca²⁺ current

Introduction

Homeostasis of blood glucose levels requires a fine‐tuned control between the two pancreatic hormones insulin and glucagon. In response to high plasma glucose concentrations, the beta‐cells of the pancreatic islets secrete insulin, which stimulates glucose uptake in fat and muscles and inhibits glucose output from the liver. At low blood glucose levels, the alpha‐cells of the islets release glucagon, which promotes hepatic glucose output. It is now widely accepted that defective pancreatic hormone secretion plays a pivotal role in the development of diabetes (Kahn et al. 2009). Most studies have focused on impaired insulin secretion. However, diabetics show excessive glucagon release at hyperglycaemia, and impaired counter‐regulatory glucagon response to hypoglycaemia (D'Alessio, 2011).

Whereas the main mechanisms involved in glucose‐stimulated insulin secretion have been well understood, much less is known about the glucose control of glucagon secretion (Gromada et al. 2007). This is to a large extent due to the difficulties of studying the relatively few alpha‐cells in the islets (10–15% in rodents). Important technical advancements have made it possible to obtain islets with fluorescent alpha‐cells (Quoix et al. 2007; Reimann et al. 2008) or to investigate alpha‐cells located in their natural environment with the use of pancreatic slices (Huang et al. 2011) with a much higher success rate concerning patch‐clamping. However, earlier studies with immunolabelling of alpha‐cells have provided substantial insight into the physiology of both isolated cells and alpha‐cells located on the surface of islets (Göpel et al. 2000). These studies have revealed that alpha‐cells possess many of the components involved in the signalling cascade underlying glucose‐stimulated insulin secretion (Walker et al. 2011). Thus, alpha‐cells possess glucose transporters, glucokinase, ATP‐sensitive K⁺ (KATP) channels and voltage‐gated Ca²⁺ channels as do beta‐cells. It is therefore at first glance surprising how beta‐cells can respond to glucose with increased secretion, whereas glucagon release from alpha‐cells is reduced during hyperglycaemia.

For the beta‐cells, glucose enters the cell via glucose transporters and undergoes several metabolic processes determining an increase in ATP levels. This increase closes KATP channels (Speier et al. 2005), depolarizing the cell and triggering electrical activity involving voltage‐gated calcium channels. The resulting increase in the cytosolic calcium concentration evokes exocytosis of insulin‐containing granules and release of insulin (Rorsman et al. 2012, 2014). For the alpha‐cells, two major, non‐exclusive explanations of the glucose‐induced inhibition of glucagon secretion have been proposed. One suggests that glucose has a direct effect on the alpha‐cells, the other that paracrine factors released from neighbouring islet cells (such as insulin, zinc or GABA from beta‐cells or somatostatin from delta‐cells) inhibit alpha‐cells (Gromada et al. 2007; Walker et al. 2011). However, experiments show that glucagon secretion is inhibited below the glucose threshold for insulin (and zinc and GABA) release or in the presence of somatostatin receptor antagonists (Vieira et al. 2007; MacDonald et al. 2007; Walker et al. 2011). These findings suggest an important role for direct, intrinsic regulation of glucagon at low glucose concentrations (in the range 1–6 mm), and two alternative theories have been proposed for explaining intrinsic glucagon regulation by glucose. One proposal suggests that glucose leads to a reduced conductivity of KATP channels resulting in a slight membrane depolarization, which prevents full action potential generation needed for complete activation of the high voltage‐activated (HVA) Ca²⁺ channel involved in exocytosis (Göpel et al. 2000; MacDonald et al. 2007; Walker et al. 2011; Zhang et al. 2013). The other proposal (Liu et al. 2004; Vieira et al. 2007) is that glucose stimulates Ca²⁺ uptake into the endoplasmic reticulum (ER), which terminates a store‐operated current leading to membrane hyperpolarization (Liu et al. 2004; Vieira et al. 2007).

Exocytosis in mouse alpha‐cells is SNARE‐protein dependent (Andersson et al. 2011), and mainly triggered by the calcium sensor synaptotagmin‐7 (Gustavsson et al. 2009) in response to Ca²⁺ influx through HVA Ca²⁺ channels (i.e. P/Q‐ and L‐type channels) (Gromada et al. 1997; Vignali et al. 2006; MacDonald et al. 2007; Gustavsson et al. 2009; De Marinis et al. 2010; Rorsman et al. 2012, 2014; Zhang et al. 2013). We note briefly that whereas earlier works (Gromada et al. 1997) suggested the involvement of ω‐conotoxin‐sensitive N‐type channels, more recent studies showed that ω‐conotoxin has unspecific effects, and that P/Q‐type Ca²⁺ channels are the major non‐L‐type HVA Ca²⁺ channel type (Vignali et al. 2006; Rorsman et al. 2012, 2014). It is of interest that glucose promotes translocation of granules to the membrane, but internalizes SNARE proteins such as SNAP‐25 leading to reduced exocytosis capacity (Andersson et al. 2011). Glucagon secretion is also influenced by neuronal and hormone control, e.g. by glucagon‐like peptide 1 (GLP‐1) and adrenaline. Indeed, glucagon‐containing granules in mice are under complex and fine‐tuned control by cAMP, which at low concentrations, as in the presence of GLP‐1, via protein kinase A (PKA) inhibits HVA calcium channels and exocytosis, but at higher concentrations, as in the presence of adrenalin, via Epac2 (exchange protein directly activated by cAMP isoform 2) mobilizes or primes granules at the L‐type calcium channels and stimulates the L‐type Ca²⁺ current (Gromada et al. 1997; De Marinis et al. 2010). Therefore, GLP‐1 and adrenalin, although they both act through cAMP, have opposite effects on glucagon secretion due to a refined and complex mechanism involving Ca²⁺ currents, granule availability and possibly changes in action potential shape, and hence a coupled model for electrical activity and exocytosis would be helpful to understand better the fine‐tuned control underlying alpha‐cell glucagon secretion.

In contrast to the long tradition of theoretical and computational investigations of electrical activity, exocytosis and other aspects of beta‐cell function (Bertram et al. 2007; Pedersen et al. 2011), very little modelling has been performed for alpha‐cells. Diderichsen & Göpel (2006) presented a mathematical model of electrical activity based on ion channel characteristics of alpha‐cells located on the surface of intact mouse islets. Very recently, Watts & Sherman (2014) presented an updated version of the model presented by Diderichsen & Göpel (2006), by including Ca²⁺ dynamics and secretion. Similarly, Fridlyand & Philipson (2012) presented theoretical studies of electrical activity, Ca²⁺ dynamics and secretion. In these studies, the main focus was on the regulation of alpha‐cell electrical activity (Diderichsen & Göpel, 2006; Fridlyand & Philipson, 2012; Watts & Sherman, 2014), whereas exocytosis and secretion were mainly a read‐out of electrical activity.

In this paper, we devise a detailed model of Ca²⁺ dynamics and exocytosis in alpha‐cells downstream of electrical activity based on recent experimental data (De Marinis et al. 2010; Zhang et al. 2013). Our focus is not on the regulation of electrical activity, which we simulate by a slightly modified version of the model presented by Watts & Sherman (2014). Rather, we characterize the intracellular Ca²⁺ dynamics with particular attention to the modelling of microdomain Ca²⁺ concentrations surrounding the P/Q‐ and L‐type Ca²⁺ channels involved in glucagon release. Our model allows us to reproduce different and independent experimental data, such as the effects of glucose and hormones (GLP‐1 and adrenaline) on glucagon secretion, and to increase our understanding of the main control mechanisms operating in the alpha‐cells.

Methods

The proposed model aims to investigate the coupling between electrical activity and exocytosis in mouse alpha‐cells based on published experimental data (De Marinis et al. 2010; Zhang et al. 2013), with focus on effects downstream of electrical activity. In particular, the model can reproduce the effects of GLP‐1 and adrenaline as well as those of glucose on alpha‐cell electrical activity and glucagon secretion. The devised model is a modified version of previous ones (Diderichsen & Göpel, 2006; Watts & Sherman, 2014). To reproduce experimental membrane potential traces as well as possible, we added minor changes to the model by Watts and Sherman (2014) as explained below. The main contribution of our work is the detailed modelling of Ca²⁺ dynamics and exocytosis.

The alpha‐cell electrical activity is described by the following equation:

$$\begin{array}{ccc}\hfill \frac{\mathrm{d}V}{\mathrm{d}t}& =& -(I_{\mathrm{CaL}}+I_{\mathrm{CaP}/\mathrm{Q}}+I_{\mathrm{CaT}}+I_{\mathrm{Na}}+I_{\mathrm{K}}+I_{\mathrm{KATP}}\hfill \\ & & +I_{\mathrm{KA}}+I_{\mathrm{L}}+I_{\mathrm{SOC}})/C_{\mathrm{m}},\hfill \end{array}$$ (1)

where V is membrane potential, I _CaL, I _CaP/Q and I _CaT are L‐, P/Q‐ and T‐type voltage‐dependent Ca²⁺ currents, respectively, I _Na is a voltage‐dependent Na⁺ current; I _K is a delayed rectifier K⁺ current, I _KA is an A‐type voltage‐dependent K⁺ current; I _KATP is an ATP‐sensitive K⁺ current, I _L is a leak current, I _SOC is a store‐operated Ca²⁺ current (SOC), and C _m is the membrane capacitance set to 5 pF. For all the currents, we exploit the same equations and parameters presented in the model devised by (Watts & Sherman, 2014) with few parameter changes (see Supporting Information including the equations for all the currents defined in eqn (1) and the corresponding Table S1 for the parameter values); however, with respect to their model, we make the assumption that the SOC conductance, g _SOC, is constant, and we rename the N‐type as the P/Q‐type voltage‐dependent Ca²⁺ currents. Indeed, as reported in Rorsman et al. (2012, 2014), mouse alpha‐cells contain low‐threshold T‐type Ca²⁺ channels and two types of HVA Ca²⁺ channels: the L‐type Ca²⁺ channels and, according to recent experiments, the P/Q‐ rather than N‐type Ca²⁺ channels. The involvement of N‐type Ca²⁺currents suggested by earlier experiments (Gromada et al. 1997) using ω‐conotoxin are in contrast to more recent findings showing that N‐type channels are very poorly expressed in alpha‐cells, and that ω‐conotoxin has unspecific effects also on P/Q‐type currents (Vignali et al. 2006; Rorsman et al. 2012, 2014). Indeed, the P/Q‐type specific blocker ω‐agatoxin inhibits alpha‐cell exocytosis and secretion strongly (Zhang et al. 2013). For the I _CaP/Q current, we exploit a similar expression used by Watts & Sherman (2014) for the I _CaN current (see appendix in Watts & Sherman, 2014), but we modify the voltage channel activation by taking into account the voltage dependence of exocytosis shown in Zhang et al. (2013), assuming that exocytosis under control conditions is completely due to P/Q‐type Ca²⁺ channel opening. The steady state activation curve of the P/Q‐type Ca² channels, $m_{\mathrm{CaP}/\mathrm{Q},\infty }$, is represented by the Boltzmann function

$$m_{\mathrm{CaP}/\mathrm{Q},\infty }\left(V\right)=\frac{1}{1+exp\left(-\frac{V-V_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}}{S_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}}\right)},$$ (2)

where $V_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}$ and $S_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}$ are set to −1 and 4 mV, respectively.

Glucagon secretion depends on Ca²⁺ dynamics; in particular, we model exocytosis as depending on Ca²⁺ concentrations in the microdomains surrounding the P/Q‐ and L‐type Ca²⁺ channels, Ca _P/Q and Ca _L respectively, and the sub‐membrane Ca²⁺ concentration, Ca _m. Moreover, we also consider the bulk cytosolic, Ca _c, and the ER, Ca _er, Ca²⁺ concentrations, in order to characterize the complete Ca²⁺ dynamics.

The following equations describe Ca _P/Q and Ca _L dynamics in single microdomains as

$$\frac{\mathrm{d}C{a}_{\mathrm{P}/\mathrm{Q}}}{\mathrm{d}t}=-f\alpha \frac{i_{\mathrm{CaP}/\mathrm{Q}}}{Vo{l}_{\mu \mathrm{d}}}-f{B}_{\mu \mathrm{d}}\left(C{a}_{\mathrm{P}/\mathrm{Q}}-C{a}_{\mathrm{m}}\right),$$ (3)

$$\frac{\mathrm{d}C{a}_{\mu \mathrm{dL}}}{\mathrm{d}t}=-f\alpha \frac{i_{\mathrm{CaL}}}{Vo{l}_{\mu \mathrm{d}}}-f{B}_{\mu \mathrm{d}}\left(C{a}_{\mathrm{L}}-C{a}_{\mathrm{m}}\right),$$ (4)

where f = 0.01 is the ratio of free‐to‐total Ca²⁺, α = 5.18×10⁻¹⁵ μmol pA^–1 ms^–1 converts current to flux, i _CaP/Q and i _CaL are the Ca²⁺ currents (possibly zero) of, respectively, P/Q‐ and L‐type entering the single domain, and Vol _μd is the volume of a single microdomain. We set the radius of the microdomains to 50 nm, the radius of the cell to 5.3 μm (the whole cell volume is equal to 0.624 pL = 624 μm³ as in Watts & Sherman, 2014), and assume a number of N _P/Q  = 100 Ca²⁺ channels for the P/Q‐type and N _L  = 400 for the L‐type (the L‐type Ca²⁺ channels outnumber non‐L‐type Ca²⁺ channels by a factor of four; Rorsman et al. 2012). B _μd describes the flux from the microdomains to the sub‐membrane, and is given as (De Schutter & Smolen, 1998):

$$B_{\mu \mathrm{d}}=D_{\mathrm{Ca}}\frac{A_{\mu \mathrm{d}}}{Vo{l}_{\mu \mathrm{d}}{d}_{\mu \mathrm{d}}},$$ (5)

where D _Ca  = 0.220 μm² ms⁻¹ is the diffusion constant for Ca²⁺, A _μd is the surface area of the microdomains and d _μd is a typical length constant of the microdomain, which is set equal to the radius of the microdomains, yielding B _μd  = 264 ms⁻¹.

Since the dynamics for Ca _P/Q and Ca _L are very fast with respect to the other variables of the system (time constant τ = 1/B _μd  = 3.8 μs) we assume steady state approximation in eqns (3) and (4). Therefore, the microdomain Ca²⁺ concentrations near open P/Q‐ and L‐type channels, Ca _P/Qo and Ca _Lo, can be described as

$$C{a_{\mathrm{P}/\mathrm{Q}}}_{\mathrm{o}}=C{a}_{\mathrm{m}}-\frac{\alpha i_{\mathrm{CaP}/\mathrm{Q}}}{B_{\mu \mathrm{d}}Vo{l}_{\mu \mathrm{d}}},$$ (6)

$$C{a_{\mathrm{L}}}_{\mathrm{o}}=C{a}_{\mathrm{m}}-\frac{\alpha i_{\mathrm{CaL}}}{B_{\mu \mathrm{d}}Vo{l}_{\mu \mathrm{d}}}.$$ (7)

Assuming no channel clustering, i _CaP/Q and i _CaL are here the single‐channel P/Q‐ and L‐type Ca²⁺ currents.

The microdomain Ca²⁺ concentrations surrounding P/Q‐ and L‐type Ca²⁺ channels when the channels close, Ca _P/Qc and Ca _Lc, respectively, are then

$$C{a_{\mathrm{P}/\mathrm{Q}}}_{\mathrm{c}}\approx C{a}_{\mathrm{m}},$$ (8)

$$C{a_{\mathrm{L}}}_{\mathrm{c}}\approx C{a}_{\mathrm{m}}.$$ (9)

The sub‐membrane Ca²⁺ concentration, Ca _m, is modelled as

$$\begin{array}{ccc}\hfill \frac{\mathrm{d}C{a}_{\mathrm{m}}}{\mathrm{d}t}& =& f\left(\frac{-\alpha I_{\mathrm{CaT}}}{Vo{l}_{\mathrm{m}}}+\frac{N_{\mathrm{P}/\mathrm{Q}}Vo{l}_{\mu \mathrm{d}}}{Vo{l}_{\mathrm{m}}}{B}_{\mu \mathrm{d}}{m}_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}\right.\hfill \\ & & \left.\left(C{a_{\mathrm{P}/\mathrm{Q}}}_{\mathrm{o}}-C_{\mathrm{m}}\right)+\frac{N_{\mathrm{L}}Vo{l}_{\mu \mathrm{d}}}{Vo{l}_{\mathrm{m}}}{B}_{\mu \mathrm{d}}{m}_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}^2{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}\left(C{a_{\mathrm{L}}}_{\mathrm{o}}-C_{\mathrm{m}}\right)\right)\hfill \\ & & -f\left(\frac{Vo{l}_{\mathrm{c}}}{Vo{l}_{\mathrm{m}}}{k}_{\mathrm{PMCA}}C{a}_{\mathrm{m}}+\frac{Vo{l}_{\mathrm{c}}}{Vo{l}_{\mathrm{m}}}{B}_{\mathrm{m}}\left(C{a}_{\mathrm{m}}-C{a}_{\mathrm{c}}\right)\right)\hfill \end{array}$$ (10)

where Vol _m and Vol _c are the volumes of the sub‐membrane compartment and the bulk cytosol, respectively, the terms $m_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}$ and $m_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}^2{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}$ represent the probability for the P/Q‐ and the L‐type Ca²⁺ channels to be opened, respectively (see Supporting Information for their definition), and k _PMCA is the rate of Ca²⁺‐ATPases through the plasma membrane and equal to 0.3 ms⁻¹. We set the sub‐membrane depth to 150 nm. B _m describes the flux from the sub‐membrane compartment to the bulk cytosol and is defined as (De Schutter & Smolen, 1998)

$$B_{\mathrm{m}}=D_{\mathrm{Ca}}\frac{A_{\mathrm{m}}}{Vo{l}_{\mathrm{c}}{d}_{\mathrm{m}}},$$ (11)

where A _m is the internal surface area of the sub‐membrane compartment. By setting d _m equal to 1 μm, B _m  = 0.128 ms⁻¹.

The bulk cytosolic Ca²⁺ concentration, Ca _c, is modelled as

$$\begin{array}{ccc}\hfill \frac{\mathrm{d}C{a}_{\mathrm{c}}}{\mathrm{d}t}& =& f\left(B_{\mathrm{m}}\left(C{a}_{\mathrm{m}}-C{a}_{\mathrm{c}}\right)+p_{\mathrm{leak}}\left(C{a}_{\mathrm{er}}-C{a}_{\mathrm{c}}\right)\right.\hfill \\ & & \left.-k_{\mathrm{SERCA}}C{a}_{\mathrm{c}}\right),\hfill \end{array}$$ (12)

where p _leak is the rate of the leak from ER to the cytosol and equal to 3×10⁻⁴ ms⁻¹, k _SERCA the rate of sarco/endoplasmic Ca²⁺‐ATPase (SERCA) pump‐dependent sequestration of Ca²⁺ into the ER and equal to 0.1 ms⁻¹.

The ER Ca²⁺ concentration, Ca _er, is modelled as

$$\frac{\mathrm{d}C{a}_{\mathrm{er}}}{\mathrm{d}t}=-f\frac{Vo{l}_{\mathrm{c}}}{Vo{l}_{\mathrm{er}}}\left(p_{\mathrm{leak}}\left(C{a}_{\mathrm{er}}-C{a}_{\mathrm{c}}\right)-k_{\mathrm{SERCA}}C{a}_{\mathrm{c}}\right),$$ (13)

where Vol _er is the volume of the ER (Vol _c /Vol _er  = 31).

The total amount of glucagon secretion, GS, is given as

$$GS\left(t\right)=G{S}_{\mathrm{P}/\mathrm{Q}}\left(t\right)+G{S}_{\mathrm{L}}\left(t\right)+G{S}_{\mathrm{m}}\left(t\right).$$ (14)

The term GS _P/Q represents the cumulative P/Q‐type microdomain‐dependent exocytosis and is modelled as

$$\begin{array}{ccc}\hfill G{S}_{\mathrm{P}/\mathrm{Q}}\left(t\right)& =& \underset{0}{\overset{t}{\int }}\left(m_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{f}_{\mathrm{H}}\left(C{a}_{\mathrm{P}/{\mathrm{Q}}_{\mathrm{o}}},K_{\mathrm{P}/\mathrm{Q}},n_{\mathrm{P}/\mathrm{Q}}\right)\right.\hfill \\ & & \left.+\left(1-m_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}\right)f_{\mathrm{H}}\left(C{a}_{\mathrm{P}/{\mathrm{Q}}_{\mathrm{c}}},K_{\mathrm{P}/\mathrm{Q}},n_{\mathrm{P}/\mathrm{Q}}\right)\right)\mathrm{d}\tau \cong \hfill \\ & & \underset{0}{\overset{t}{\int }}\left(m_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{f}_{\mathrm{H}}\left(C{a}_{\mathrm{P}/{\mathrm{Q}}_{\mathrm{o}}},K_{\mathrm{P}/\mathrm{Q}},n_{\mathrm{P}/\mathrm{Q}}\right)\right.\hfill \\ & & \left.+\left(1-m_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}\right)f_{\mathrm{H}}\left(C{a}_{\mathrm{m}},K_{\mathrm{P}/\mathrm{Q}},n_{\mathrm{P}/\mathrm{Q}}\right)\right)\mathrm{d}\tau \hfill \end{array}$$ (15)

where, as in Pedersen & Sherman (2009),

$$f_{\mathrm{H}}\left(x,K,n\right)=\frac{x^n}{x^n+K^n}.$$ (16)

The nominal values of n _P/Q and K _P/Q are 4 and 2 μm, respectively. Note that the first term of the integral represents the secretion depending on Ca _P/Qo, while the second term the secretion depending on Ca _P/Qc ($1-m_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}$ is the fraction of closed P/Q‐type Ca²⁺ channels).

The term GS _L models the total glucagon release from L‐type microdomains and follows

$$\begin{array}{ccc}\hfill G{S}_{\mathrm{L}}\left(t\right)& =& \underset{0}{\overset{t}{\int }}\left(m_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}^2{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}{f}_{\mathrm{H}}\left(C{a}_{{\mathrm{L}}_{\mathrm{o}}},K_{\mathrm{L}},n_{\mathrm{L}}\right)\right.\hfill \\ & & +\left.\left(1-m_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}^2{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}\right)f_{\mathrm{H}}\left(C{a}_{{\mathrm{L}}_{\mathrm{c}}},K_{\mathrm{L}},n_{\mathrm{L}}\right)\right)\mathrm{d}\tau \cong \hfill \\ & & \underset{0}{\overset{t}{\int }}\left(m_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}^2{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}{f}_{\mathrm{H}}\left(C{a}_{{\mathrm{L}}_{\mathrm{o}}},K_{\mathrm{L}},n_{\mathrm{L}}\right)\right.\hfill \\ & & +\left.\left(1-m_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}^2{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}\right)f_{\mathrm{H}}\left(C{a}_{\mathrm{m}},K_{\mathrm{L}},n_{\mathrm{L}}\right)\right)\mathrm{d}\tau \hfill \end{array}$$ (17)

We set the nominal values of n _L and K _L to 4 and 50 μm, respectively. The low‐affinity assumption K _L = 50 μm reflects that glucagon secretion in the basal state is independent of L‐type Ca²⁺ currents (Gromada et al. 1997; De Marinis et al. 2010). Similarly to eqn (15), the first term of the integral represents the secretion depending on Ca _Lo, while the second term the secretion depending on Ca _Lc ($1-m_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}^2{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}$ is the probability that an L‐type Ca²⁺ channel is closed).

Finally, cumulative secretion depending on the sub‐membrane Ca²⁺ concentration is given as

$$G{S}_{\mathrm{m}}\left(t\right)=\underset{0}{\overset{t}{\int }}{f}_{\mathrm{H}}\left(C{a}_{\mathrm{m}},K_{\mathrm{m}},n_{\mathrm{m}}\right)\mathrm{d}\tau .$$ (18)

As for GS _P/Q, we set n _m and K _m to 4 and 2 μm, respectively. The K _P/Q and K _m values are based on the results that exocytosis rises when the intracellular concentration Ca²⁺ reaches the threshold value of 2 μm (Huang et al. 2011).

The glucagon secretion rate due to the different compartments (GSR _P/Q, GSR _Land GSR _m) plotted in the figures is calculated by approximating the derivative of the corresponding cumulative secretion with the difference quotient in time windows of 15 s.

Modelling glucose effects

We simulate the effect produced by glucose increase by decreasing the KATP‐channel conductance, g _KATP (Gromada et al. 2004; Zhang et al. 2013), and increasing the leak conductance, g _L. We set the g _KATP nominal value (i.e. in the presence of 1 mm glucose) to 0.3 nS and g _L nominal value to 0.2 nS. Then, we reproduce the effect produced by 6 mm glucose by decreasing g _KATP to 0.2475 nS and increasing g _L to 0.35 nS linearly over time. All the other parameters are not modified and fixed to their nominal values. The increased leak conductance is speculative, but allows us to reproduce the experimentally observed electrical patterns better, which in turn permits us to study the mechanisms downstream of changes in electrical activity, rather than the mechanisms leading to these changes.

Modelling GLP‐1 and adrenaline effects

We model the inhibition effect of GLP‐1 by varying the parameter $V_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}$ of the sigmoidal function defined by eqn (2), describing the steady state activation curve of the P/Q‐type Ca²⁺ channels. We increase $V_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}$ from its nominal value (−1 mV) to 30 mV, reproducing the strong inhibition of the P/Q‐type Ca²⁺ channels via PKA, which is activated through a small increase of cAMP produced by GLP‐1 (De Marinis et al. 2010). In addition, we decrease the leak conductance, g _L, from its nominal value to 0.165 nS, which allows us to reproduce the transient hyperpolarization of the cell. All the other parameters are not modified and fixed to their nominal values.

The effect of adrenaline is reproduced by increasing the L‐type Ca²⁺ channel conductance, g _CaL, and decreasing the threshold value for exocytosis near L‐type Ca²⁺ channels, K _L, in order to reproduce the Epac2‐mediated increase of the L‐type Ca²⁺ currents and possible priming or recruitment of granules around the L‐type Ca²⁺ channels (Gromada et al. 1997; De Marinis et al. 2010). Moreover, we increase the leak conductance, g _L, and the parameter $V_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}$ as for the simulated GLP‐1 effect. The increased leak conductance could represent a cAMP‐regulated chloride channel, e.g. the cystic fibrosis transmembrane conductance regulator (CFTR) (Boom et al. 2007; Edlund et al. 2014). We simulate the effects of adrenaline for two different glucose levels: at low levels (1 mm) and at physiological levels (6 mm). At low glucose levels we increase g _L to 0.37 nS linearly over time, while at physiological levels we increase g _L to 0.38 nS and decrease g _KATP to 0.2475 nS (as simulated for the glucose effects) linearly over time. For both cases, we increase g _CaL to 0.9775 nS (15% more than its nominal value, 0.85 nS), and $V_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}$ to 30 mV, while decreasing K _L to 2 μm linearly over time. All the other parameters are not modified and kept to their nominal values.

Results

The devised mathematical model of electrical activity and exocytosis in mouse alpha‐cells is an updated version of previous models (Diderichsen & Göpel, 2006; Watts & Sherman, 2014) based on recent experimental data (De Marinis et al. 2010; Zhang et al. 2013). Although alpha‐cell electrical activity is well studied (Diderichsen & Göpel, 2006; Fridlyand & Philipson, 2012; Watts & Sherman, 2014), coupling to exocytosis is not. Therefore, our model couples the electrical activity and exocytosis to give a unified picture of the main mechanisms that control glucagon secretion; we characterize the complete Ca²⁺ dynamics with particular attention to the modelling of microdomain Ca²⁺ concentrations surrounding the P/Q‐ and L‐type Ca²⁺ channels involved in glucagon release. Here, we show how the model is able to reproduce the effects of glucose increase as well as GLP‐1 and adrenaline on glucagon exocytosis.

Control glucose of glucagon secretion

Figure 1 shows the results of the devised model by reproducing the effect of the glucose increase. As explained in the Methods, we simulate the effect produced by 6 mm glucose by decreasing the KATP‐channel conductance, g _KATP, and increasing the leak conductance, g _L, between time t = 2 min and t = 3 min. We keep these conditions until time t = 6 min, then, between t = 6 min and t = 7 min, we simulate the return to the initial conditions (i.e. in the presence of 1 mm glucose) by resetting the modified model parameters to their nominal values. The model is able to reproduce electrical activity characterized by spikes with reduced amplitude due to the inhibitory effect of the glucose increase (Fig. 1 A). As found experimentally (Zhang et al. 2013), both the reduction in action potential peak voltage and the depolarization of the interspike potential are ∼10 mV. The decrease of spike amplitude determines a significant reduction in glucagon secretion, in particular the P/Q‐type Ca²⁺ channel‐dependent secretion (GS _P/Q defined by eqn (15)), is significantly reduced (see the dashed‐dotted plot in Fig. 1 B showing the glucagon secretion rate from the P/Q‐type microdomains,GSR _P/Q), because few P/Q‐type Ca²⁺ channels are activated, and thus the probability of these channels being opened is very low, as shown by the reduction of the term $m_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}$ (Fig. 1 E), resulting in decreased glucagon secretion as seen from eqn (15). Note, instead, that the microdomain Ca²⁺ concentration surrounding the open P/Q‐type Ca²⁺ channels exhibits a small decrease of the peak value and a small increase of the minimum value (Fig. 1 C) that do not affect the value of the function f_H (see eqns (15) and (16)). Thus, it is the reduction of the number of open P/Q‐type Ca²⁺ channels rather than a variation in local Ca²⁺ levels that underlies the inhibition of glucagon release by glucose in the model.

Figure 1. Model results reproducing the effect of glucose increase between time t = 2 min and t = 6 min .

Simulated electrical activity (A) and the corresponding total glucagon secretion rate, GSR (solid plot in B), given by the sum of GSR _P/Q (dashed‐dotted), GSR _L (dashed) and GSR _m (dashed with circles). GSR _P/Q is a function of the microdomain Ca²⁺ concentration surrounding the open P/Q‐type Ca²⁺ channels,Ca _P/Qo (C), the probability for the P/Q‐type Ca²⁺ channels to be opened, $m_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{P}/\mathrm{Q}}}$(E), its complement, and Ca _m (G) (see eqns (15) and (16)). GSR _L is a function of the microdomain Ca²⁺ concentration surrounding the open L‐type Ca²⁺ channels,Ca _Lo (D), the probability for the L‐type Ca²⁺ channels to be opened, $m_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}^2{h}_{\mathrm{C}{\mathrm{a}}_{\mathrm{L}}}$ (F), its complement, and Ca _m (G) (see eqns (16) and (17)). GSR _m is a function of Ca _m (G) (see eqn (18)). H, cytosolic Ca²⁺ concentration, Ca _c.

The L‐type Ca²⁺ channels are not significant for glucagon secretion (see the dashed plot in Fig. 1 B showing the glucagon secretion rate from the L‐type microdomains, GSR _L) due to our parameter choice: the L‐type Ca²⁺ channel‐dependent secretion is negligible also when the L‐type Ca²⁺ channels open; indeed, the microdomain Ca²⁺ concentration value surrounding the open L‐type Ca²⁺ channels (see Fig. 1 D) is much lower than the threshold value of the function f_H defined by the parameter K _L (see eqns (16) and (17), in particular f _H∼0 which implies that GS _L is virtually zero). Note that the reduced spike amplitude results in a small decrease of the probability for the L‐type Ca²⁺ channels to be opened (Fig. 1 F). The secretion depending on the sub‐membrane Ca²⁺ concentration decreases as the glucose level increases (see the dashed plot with circle markers in Fig. 1 B showing the glucagon secretion rate from the sub‐membrane compartment, GSR _m) because Ca _m shows a reduced peak value (Fig. 1 G). Finally, Ca _c shows a slight increase during the simulated glucose increase (Fig. 1 H), demonstrating that whole‐cell Ca²⁺ imaging does not necessarily reflect secretion in alpha‐cells (Le Marchand & Piston, 2010; Nakamura & Bryan, 2014; Rorsman et al. 2014; Elliott et al. 2015).

Effects of GLP‐1 on glucagon secretion

Figure 2 shows the results of the model reproducing the effect of GLP‐1. As explained in the Methods, we simulate this effect by increasing the parameter $V_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}$ of the steady state activation curve of the P/Q‐type Ca²⁺ channels (see eqn (2)), reflecting the GLP‐1‐mediated reduction of P/Q‐type Ca²⁺ currents (De Marinis et al. 2010), and decreasing the leak conductance g _L. The model reproduces the transient hyperpolarization of the cell and subsequently depolarization due to the simulated inhibitory effect of GLP‐1, as observed experimentally (De Marinis et al. 2010). The P/Q‐type Ca²⁺ channel‐dependent secretion, after the transient, is significantly reduced (see the dashed‐dotted plot in Fig. 2 B showing GSR _P/Q) due to the increase of $V_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}$: the P/Q‐type Ca²⁺ channels are closed (Fig. 2 E). Note, instead, that, during the transient, the minimum value of microdomain Ca²⁺ concentration surrounding the open P/Q‐type Ca²⁺ channels exhibits a small decrease (Fig. 2 C) while the maximum value a very low increase that do not affect the value of the function f _H (see eqns (15) and (16)). Thus, as above, it is the closing of the P/Q‐type Ca²⁺ channels rather than a change in local Ca²⁺ levels that underlies the inhibition of glucagon release by GLP‐1 in the model. As for the previous case, the L‐type Ca²⁺ channels are not relevant for glucagon secretion (see the dashed plot in Fig. 2 B showing GSR _L): the microdomain Ca²⁺ concentration surrounding the open L‐type Ca²⁺ channels (see Fig. 2 D) is much lower than the threshold value of the function f _H defined by the parameter K _L (i.e. f _H∼0 in eqn (17)). Note that the simulated GLP‐1 effect does not greatly change the probability for the L‐type Ca²⁺ channels to be opened (Fig. 2 F). Secretion depending on the sub‐membrane Ca²⁺ concentration decreases due to the simulated effect of GLP‐1 (see the dashed plot with circle markers in Fig. 2 B showing GSR _m), as Ca _m decreases (Fig. 2 G). Finally, Ca _c shows a slight decrease during the simulated GLP‐1 effects (Fig. 2 H).

Figure 2. Model results reproducing the effect of GLP‐1 between time t = 2 min and t = 9 min .

Panel details are as in Fig. 1.

Effects of adrenaline on glucagon secretion

Figure 3 shows the results of the model reproducing the effect of adrenaline at low (1 mm) and at physiological (6 mm) glucose levels. As explained in the Methods, we simulate this effect by increasing g _CaL, g _L and $V_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}$, and decreasing K _L. In addition, at physiological glucose levels, we also decrease g _KATP, as simulated for the glucose effects (Fig. 1). The reduction in K _L corresponds to either priming of granules located at L‐type channels or to recruitment of granules to the L‐type channels (Gromada et al. 1997), two hypothesized Epac2‐mediated processes that become active at high cAMP concentrations (De Marinis et al. 2010).

Figure 3. Model results reproducing the effect of adrenaline between time t = 2 min and t = 6 min .

Panel details are as in Fig. 1. The continuous lines are the results obtained at low glucose levels (1 mm), while the grey dotted lines are the results at physiological glucose levels (6 mm).

At low glucose concentration, the simulated electrical activity exhibits reduced spike amplitude in response to adrenaline (Fig. 3 A), with both the action potential peak voltage and the interspike potential modified by ∼10 mV, as shown experimentally (De Marinis et al. 2010). The effect of higher glucose levels determines an additional decrease of spike amplitude (see the grey dotted plot in Fig. 3 A). The L‐type Ca²⁺ channels mainly determine the glucagon secretion increase (see the dashed plot in Fig. 3 B showing GSR _L) due to adrenaline: the microdomain Ca²⁺ concentration surrounding the open L‐type Ca²⁺ channels (Fig. 3 D) increases at both glucose levels and is much higher than the threshold value K _L, the value of which is decreased considerably. This effect compensates for the small decrease of the probability for the L‐type Ca²⁺ channels to be opened (Fig. 3 F). The P/Q‐type Ca²⁺ channel‐dependent secretion is virtually absent (see the dashed‐dotted plot in Fig. 3 B showing GSR _P/Q) due to the increase of $V_{{\mathrm{m}}_{\mathrm{CaP}/\mathrm{Q}}}$, determining the closure of the P/Q‐type Ca²⁺ channels (Fig. 3 E), as shown for the simulated GLP‐1 effect. Note, instead, that the microdomain Ca²⁺ concentration surrounding the open P/Q‐type Ca²⁺ channels exhibits a small decrease of the peak value and a small increase of the minimum value for both glucose levels (Fig. 3 C) that do not affect the value of the function f _H (see eqns (15) and (16)). The secretion depending on the sub‐membrane Ca²⁺ concentration decreases slightly (see the dashed plot with circle markers in Fig. 3 B showing GSR _m), as Ca _m exhibits a reduced peak value for both cases (Fig. 3 G). Instead, the intracellular Ca²⁺ concentration, Ca _c, shows a small increase for both glucose levels (Fig. 3 H). In summary, our model reproduces both in vitro data, but at low, non‐physiological glucose concentration (De Marinis et al. 2010), and the physiological increase in glucagon release during exercise or flight‐or‐fight response (Galbo et al. 1975; Wasserman et al. 1984; Hirsch et al. 1991; Marker et al. 1991; Vieira et al. 2004; Jones et al. 2012), when both glucose and adrenalin are high.

Discussion

Alpha‐cells and, in particular, the main mechanisms underlying glucagon release represent a complex system that, when defective, is involved in the development of diabetes (D'Alessio, 2011). Therefore, alpha‐cells are being considered as a possible pharmacological target for the treatment of diabetes (Christensen et al. 2011) and, for this reason, a better understanding of the regulation of glucagon secretion by glucose and other physiological factors is needed.

Here we addressed this aim by developing a mathematical model that couples electrical activity and exocytosis to achieve a full picture of the role of the mechanisms underlying glucagon release. Our main focus was not the regulation of electrical activity, which has been well studied recently (Diderichsen & Göpel, 2006; Fridlyand & Philipson, 2012; Watts & Sherman, 2014), but rather the main mechanisms downstream of electrical activity that determine the regulation of glucagon secretion. Thus, we devised a detailed model of Ca²⁺ dynamics by exploiting compartmental modelling of Ca²⁺ levels in the microdomains surrounding the HVA Ca²⁺ channels (P/Q‐ and L‐type Ca²⁺ channels), in the sub‐membrane Ca²⁺ compartment, in the bulk cytosol and in the ER, to provide a deeper knowledge of the relative contributions of the various subcellular Ca²⁺ compartments in the control of glucagon secretion.

The model reproduced the effects of glucose increase as well as GLP‐1 and adrenaline on electrical activity and glucagon secretion, by simulating changes of ion channel characteristics (i.e. conductance, inactivation and activation functions), granule availability, and putative priming or recruitment of granules around the HVA Ca²⁺ channels. Most of the changes are experimentally supported, while others are speculative in order to reproduce the experimental data and could guide new experiments to validate our hypotheses.

In agreement with experiments (Zhang et al. 2013), simulated glucose increase caused a moderate depolarization with decreased action potential height (see Fig. 1 A) and resulting reduction of P/Q‐type Ca²⁺ channel activation (Fig. 1 E) that lead to decreased glucagon secretion (Fig. 1 B). Therefore, the inhibition of glucagon release by glucose increase is mainly determined by the reduction of the number of open P/Q‐type Ca²⁺ channels, rather than a variation in local calcium levels, which were slightly raised by glucose.

Moreover, the model reproduced paracrine effects such as inhibition of glucagon secretion by GLP‐1 or stimulation of release by adrenaline. Both GLP‐1 and adrenaline act through cAMP but with different effects. On the one hand, GLP‐1 causes a small increase in intracellular cAMP concentration that activates PKA, leading to inactivation of the P/Q‐type Ca²⁺ channels (De Marinis et al. 2010). On the other hand, adrenaline produces a large increase in intracellular cAMP concentration, leading not only to inhibition of the P/Q type Ca²⁺ channels, as for GLP‐1, but also to activation of the low‐affinity cAMP sensor Epac2, which causes a rise of the L‐type Ca²⁺ currents and the possible priming or recruitment of granules in the microdomains around the L‐type Ca²⁺ channels (De Marinis et al. 2010). Recently, it was shown that these effects depend on SUMOylation (Dai et al. 2014) (SUMO, small ubiquitin‐like modifier protein).

We reproduced the GLP‐1 effects by assuming closure of the P/Q type Ca²⁺ channels (Fig. 2 E), and therefore in the model it is the closing of these channels that causes the strong inhibition of glucagon secretion (Fig. 2 B). The model simulated the stimulation of glucagon release produced by adrenaline (Fig. 3 B) through a rise of the L‐type Ca²⁺ channel currents combined with an increase of the granule affinity for the Ca²⁺ levels in the microdomains around the L‐type Ca²⁺ channels, reflecting the possible priming or recruitment of granules in these microdomains.

For the simulated effects produced by glucose increase and adrenaline, the model showed a rise of the bulk cytosolic calcium levels Ca _c during the simulated effects (Figs 1 H and 3 H). However, only for adrenaline was an increase of glucagon secretion correlated with a rise in Ca _c. In this context it is worth noting that alpha‐cell Ca²⁺ levels in response to elevations in glucose concentrations have been reported to increase (Le Marchand & Piston, 2010; Rorsman et al. 2014) or remain unchanged (Nakamura & Bryan, 2014; Elliott et al. 2015), while other works have found a slight decrease (MacDonald et al. 2007; Quoix et al. 2009), but sometimes only in a fraction of the cells (Salehi et al. 2006). This discrepancy suggests that whole‐cell Ca²⁺ imaging does not necessarily reflect secretion in alpha‐cells, as shown by our simulations. In particular, our model highlights how glucose uncouples the positive relationship between the cytosolic Ca²⁺ concentration and secretory activity (Le Marchand & Piston, 2010, 2012) because of control of exocytosis by localized Ca²⁺ elevations. Furthermore, note that the model exhibited a similar behaviour in terms of electrical activity for the simulated effects produced by glucose increase (Fig. 1 A) and adrenaline (Fig. 3 A), as observed experimentally (De Marinis et al. 2010; Zhang et al. 2013). However, we showed how the various mechanisms involved in exocytosis downstream of electrical activity determine different glucagon release responses.

The exact molecular mechanisms controlling the local events involving calcium channel opening and granule availability in alpha‐cells are to a large extent unknown. SUMOylation has been suggested to mediate the Epac2‐dependent increase in the L‐type Ca²⁺ channel density involved in adrenalin‐stimulated glucagon secretion simulated in Fig. 3(Dai et al. 2014). Another possibility could be regulation by L‐type Ca²⁺ channel trafficking to and from the cell membrane as in insulin‐secreting cells (Buda et al. 2013). It may be speculated that this increment will facilitate formation of Ca²⁺ channel clusters able to trigger exocytosis. Other mechanisms act via electrical activity. For example, glucose‐dependent inhibition of KATP channels lowers the action potential height, thus preventing P/Q‐type channel opening (Fig. 1) (Zhang et al. 2013). GLP‐1 inhibits P/Q‐type Ca²⁺ currents via a PKA‐dependent but unknown mechanism. We modelled the reduction in P/Q‐type Ca²⁺ current as a rightshift of the activation curve (Fig. 2), but it might also be speculated that a reduced number of P/Q‐type Ca²⁺ channels at the cell membrane, as has been found for L‐type channels in beta‐cells in response to glucose (Buda et al. 2013), might underlie the reduction in the number of open P/Q‐type Ca²⁺ channels, and GLP‐1‐mediated inhibition of glucagon release.

A minor drawback of our studies is that they are based on non‐physiological experiments, investigating very low glucose concentration, and isolated effects of GLP‐1 or adrenaline (Gromada et al. 1997; De Marinis et al. 2010). The advantage is that these extreme conditions allow us to separate otherwise masked mechanisms. Using the model we were then able to simulate the combined and more physiological response to elevated glucose concentrations and adrenaline (Fig. 3), a condition that mimics the in vivo condition during exercise or flight‐or‐fight responses (Galbo et al. 1975; Wasserman et al. 1984; Hirsch et al. 1991; Marker et al. 1991; Vieira et al. 2004; Jones et al. 2012). Our simulations reproduced the well‐known increase in glucagon secretion (Luyckx & Lefebvre, 1974; Galbo et al. 1975; Wasserman et al. 1989; Ramnanan et al. 2011; Jones et al. 2012), but in addition gave mechanistic insight. The increase in release from L‐type Ca²⁺ microdomains (the adrenalin effect) is predicted to be sufficient to overcome the reduced action potential peak amplitude (the glucose effect). Other physiological scenarios such as the effects of free fatty acids (FFAs) or amino acids can, in our opinion, also be better understood within the theoretical framework presented here, once the basic mechanisms are discovered. For example, the stimulation of glucagon release by the FFA palmitate during short‐term exposure is mainly due to enhanced Ca²⁺ entry via L‐type Ca²⁺ channels (Olofsson et al. 2004 a), but also because of involvement of additional mechanisms having a direct effect on exocytosis as shown for beta‐cells (i.e. increasing the size of the readily releasable pool of granules)(Olofsson et al. 2004 b, 2004 a), similarly to the assumptions of the adrenaline simulation (Fig. 3). Thus, FFAs are predicted to increase glucagon release similarly to adrenaline.

Divergent effects on glucagon secretion can take place depending on the type and concentration of amino acids: arginine, alanine and glutamine exhibit stimulatory effects, while lysine inhibits the secretion; leucine has stimulatory effects at physiological concentrations and inhibitory at elevated levels (Quesada et al. 2008; Marroquí et al. 2014). However, little is known about the molecular mechanisms involved in the effects of most amino acids on glucagon secretion. One study reports that glutamine antagonizes glucose metabolism and closure of KATP channels in alpha‐cells (Ostenson & Grebing, 1985). The model predicts that the simulated reduction in action potential height in response to glucose (Fig. 1), which is the result of KATP‐channel closure, and the corresponding downstream inhibitory effect on glucagon release, due to the reduction of the number of open P/Q‐type Ca²⁺ channels, would be attenuated in the presence of glutamine. This can explain how glucagon secretion at high glucose is higher in the presence of glutamine (Greenfield et al. 2009). Other amino acids such as arginine may be involved in the stimulation of glucagon release by membrane depolarization and increased calcium influx (Marroquí et al. 2014), similarly to the assumptions for the adrenaline simulation (Fig. 3).

We believe that mathematical modelling is an indispensable tool to handle the delicate and complex system of the pancreatic alpha‐cell, taking our understanding of glucagon secretion a step forward, and the theoretical framework presented here is an example of the strength of modelling. The insight obtained from the proposed model could guide new experiments to focus on the main components of the pancreatic alpha‐cell involved in the coupling of electrical activity and exocytosis in order to (i) distinguish between alternative explanations of the data, (ii) test model predictions of possible disturbances in diabetic alpha‐cells and (iii) eventually improve the control of glucagon release by predicting promising drug targets in the alpha‐cell signalling cascade.

Additional information

Competing interests

The authors declare no competing interests.

Author contributions

F.M. and M.G.P. designed and performed the research, analysed the results and wrote the paper.

Funding

M.G.P. was supported by the University of Padova (PRAt 2012, and the Strategic Research Project 2012 ‘DYCENDI’).

Supporting information

Supporting Text S1. Supplementary text including the equations and parameters of the model.

Supporting File S1. MATLAB code for generating the results presented in the article.