Modelling of energy-driven switch for glucagon and insulin secretion

Journal of Theoretical Biology 493 (2020) 110213

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Modelling of energy-driven switch for glucagon and insulin secretion Vladimir Grubelnik a, Jan Zmazek b, Rene Markovicˇ a,b, Marko Gosak b,c, Marko Marhl b,c,d,∗ a

Faculty of Electrical Engineering and Computer Science, University of Maribor, Maribor SI-2000, Slovenia Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor SI-2000, Slovenia c Faculty of Medicine, University of Maribor, Maribor SI-2000, Slovenia d Faculty of Education, University of Maribor, Maribor SI-2000, Slovenia b

a r t i c l e

i n f o

Article history: Received 16 September 2019 Revised 19 February 2020 Accepted 24 February 2020 Available online 25 February 2020 Keywords: Diabetes Pancreatic hormones Mitochondria Glycolysis Free-fatty acids

a b s t r a c t We present a mathematical model of the energy-driven metabolic switch for glucagon and insulin secretion from pancreatic alpha and beta cells, respectively. The energy status related to hormone secretion is studied for various glucose concentrations. Additionally, the physiological response is studied with regards to the presence of other metabolites, particularly the free-fatty acids. At low glucose, the ATP production in alpha cells is high due to free-fatty acids oxidation in mitochondria, which enables glucagon secretion. When the glucose concentration is elevated above the threshold value, the glucagon secretion is switched off due to the contribution of glycolytic ATP production, representing an “anaerobic switch”. On the other hand, during hypoglycemia, the ATP production in beta cells is low, reflecting a “waiting state” for glucose as the main metabolite. When glucose is elevated above the threshold value, the oxidative fate of glucose in mitochondria is the main source of energy required for effective insulin secretion, i.e. the “aerobic switch”. Our results show the importance of well-regulated and fine-tuned energetic processes in pancreatic alpha and beta cells required for efficient hormone secretion and hence effective blood glucose regulation. These energetic processes have to be appropriately switched on and off based on the sensing of different metabolites by alpha and beta cells. Our computational results indicate that disturbances in cell energetics (e.g. mitochondrial dysfunction), and dysfunctional metabolite sensing and distribution throughout the cell might be related to pathologies such as metabolic syndrome and diabetes. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Type II Diabetes Mellitus (T2DM) is a metabolic disorder with epidemic dimensions that is characterized metabolically by insulin resistance and insulin secretory defects, which in turn leads to hyperglycemia (Weyer et al., 1999). The etiology and causes of this multifactorial disease are nowadays still insufficiently understood. Homeostasis of blood glucose is complex and involves several mechanisms. A necessary component is a fine-tuned control of various hormones, among which insulin and glucagon are of key importance (Walker et al., 2011). Insulin and glucagon are pancreatic hormones, secreted from pancreatic beta and alpha cells, respectively. In response to high plasma glucose concentrations, beta-cells of the pancreatic islets of Langerhans secrete insulin, which stimulates glucose uptake in liver, muscle and fat cells, while glucose output is inhibited. At low blood glucose levels, the

∗ Corresponding author at: Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor SI-20 0 0, Slovenia. E-mail address: [email protected] (M. Marhl).

https://doi.org/10.1016/j.jtbi.2020.110213 0022-5193/© 2020 Elsevier Ltd. All rights reserved.

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 (Göke, 2008; Henquin and Rahier, 2011). For several decades, the research focus has mainly been insulin resistance and the consequent defects in pancreatic beta cells and insulin secretion. The first clinical therapies had evolved around this concept; however, with only limited success. Therefore, it is getting even more evident that the T2DM has to be considered as a bi-hormonal defect, and glucagon excess, rather than insulin deficiency, appears to be the sine qua non of diabetes (Unger and Cherrington, 2012). To elucidate the complex interplay between insulin and glucagon secretion, explain the causes and to explore new possible treatments for T2DM, the exact biophysical and physiological mechanisms need to be found. Recently, it has been revealed that beta cells can undergo dedifferentiation, i.e. loss of beta-cell identity and obtaining alphalike properties, which could be the cause for T2DM rather than beta-cell death (Talchai et al., 2012). Previously, there have also been efforts to generate beta cells from other sources such as

2

V. Grubelnik, J. Zmazek and R. Markovicˇ et al. / Journal of Theoretical Biology 493 (2020) 110213

alpha and acinar cells (Puri et al., 2015) by mechanisms of transdifferentiation and reprogramming. Transdifferentiation between alpha and beta cell is possible because both cell types are derived from a common neurogenin3-expressing precursor cell in the late stage of fetal development (Baeyens et al., 2018). Therefore, despite the opposing physiological roles, pancreatic alpha and beta cells share many similarities. Both cell types are electrically excitable and are equipped with the same type of ATP-dependent potassium (KATP ) channels (Diao et al., 2008; Gromada et al., 2004; Rorsman et al., 2014). They also possess a common glucose-sensing mechanism via the expression of the glucokinase (hexokinase IV, GK), a high-Km enzyme which is not inhibited by its product glucose6-phosphate (G6P) (Heimberg et al., 1996), and they both share the same mechanism of calcium-mediated exocytosis of hormonecontaining granules (Rorsman and Ashcroft, 2018). There are, however, important differences between alpha and beta cells. They differ in size, metabolism, and the mechanisms of their electrical activity. At low glucose concentrations, beta cells are electrically silent (-70 mV), while alpha cells fire action potentials from approximately −55 mV (Rorsman et al., 2014). Moreover, under hypoglycemic conditions, the KATP -channel activity is very high in beta cells, and the depolarizing current is mainly due to L-type calcium channels, triggering insulin exocytosis. Conversely, KATP -channel activity in alpha cells is low (Zhang et al., 2013) and the depolarizing current consists of the predominant Na+ current and T-, L- and P/Q-type voltage-dependent calcium channels. The latter seem to be particularly important for the glucagon exocytosis (Ramracheya et al., 2010). Glucose intake is mediated mainly via glucose transporter isoforms GLUT1 and GLUT2 in alpha and beta cells, respectively (Heimberg et al., 1995). When differences in cellular volume are taken into account, the rates of glucose uptake and its utilization in both cell types are similar (Heimberg et al., 1996; Schuit et al., 1997). Despite the same level of glycolytic activity, the glucose oxidative capacity of alpha cells is much lower, only 1/3 to 1/6 of that in beta cells (Schuit et al., 1997). Several studies have further discussed that glucose metabolism is aerobic in beta cells and mainly anaerobic in alpha cells (Detimary et al., 1998; Quesada et al., 2006; Schuit et al., 1997). This anaerobic glucose metabolism is also well-grounded in experimental findings, showing low expression of lactate dehydrogenase (LDH) and MCT-1 genes in alpha cells (Schuit et al., 1997; Sekine et al., 1994; Thorrez et al., 2011). The question arises whether the anaerobic nature of alpha cells is significant (indispensable) for the physiological functioning of the alpha cells. This question is, in fact, more general. The anaerobic use of glucose is wasteful, not yielding the maximum energy per glucose molecule. This so-called “over-flow metabolism” (Vazquez, 2018) seems to be unacceptable for multicellular organisms (Bouillaud, 2009); however, the power of the metabolism providing anaerobic glycolysis is very high, and could be of advantage, e.g., in cancer cells or muscle cells under specific conditions (Vazquez, 2018). The anaerobic glycolysis does not necessarily mean that mitochondria are dysfunctional, as it has also been shown in many cancer cells, contradicting the Warburg’s original hypothesis (Weinberg and Chandel, 2015). Although the structure of pancreatic tissue shows that mitochondria are less numerous and with less developed cristae in alpha than in beta cells (Luchini et al., 2015; Machino et al., 1966; Mikami and Mutoh, 1971; Munger, 1958), it does not necessarily mean that the mitochondria in alpha cells are less efficient. Free-fatty acids (FFA), which are particularly important during hypoglycemia and are therefore preferable during fasting, undergo beta-oxidation inside the mitochondrial matrix, taking up the mitochondrial capacity (Briant et al., 2018a). For half a century, it has been known that the metabolism of FFA in alpha cells is crucial for glucagon secretion (Edwards and

Taylor, 1970; Hong et al., 2007). It depends on FFA in a time- and dose-dependent manner, as well as on the type of FFA (Gromada et al., 2007). The experiments with etomoxir explicitly show that the effect of FFA on the glucagon secretion is mediated by their oxidation (Hong et al., 2007; Kristinsson et al., 2017). The importance of FFA oxidation in alpha cells is further supported by the high expression of UCP2 channels in alpha cells, much more than in beta cells (Diao et al., 2008). The FFA are known stimulators of the expression of the UCP2 mRNA (Bouillaud, 2009; Millet et al., 1997; Schrauwen et al., 2001). However, despite the FFA oxidation, glucose is also indispensable for glucagon secretion (Hong et al., 2007). It has been shown that glucokinase in alpha cells is required for glucose sensing and regulation of glucagon secretion in alpha cells. In particular, it has been predicted that the glucokinase expression is a prerequisite for suppressing glucagon release during high glucose concentrations (Heimberg et al., 1996). Taken together, if the glucokinase expression in alpha cells is necessary for the glucose-induced suppression of glucagon release (Heimberg et al., 1996), and at the same time the glucose fate is mainly anaerobic (Detimary et al., 1998; Quesada et al., 2006; Schuit et al., 1997), the glycolysis appears to be an indispensable part of the glucagon-turn-off mechanism. The energy provided by the glycolysis, i.e. the anaerobic part of the glucose fate, seems to be important for the energy-driven switch suppressing glucagon secretion in healthy alpha cells. In this paper, we focus on analyzing the mechanism of these energy-driven processes in alpha and beta cells. In the current literature, there is a lack of bioenergetic analyses for alpha cells, whereas for beta cells more data about these mechanisms exists (Affourtit et al., 2018; Nicholls, 2016). For both, alpha and beta cells, it is well accepted that energy is crucial for the processes of hormone secretion. The most studied energy-dependent mechanism for the regulation of hormonesecretion in alpha and beta cells are the plasma membrane KATP channels (Briant et al., 2017; MacDonald et al., 2007; Östenson et al., 1980; Quesada et al., 2006; Zhang et al., 2013). The KATP channels are identical in alpha and beta cells (Diao et al., 2008; Gromada et al., 2004; Olsen et al., 2005; Rorsman et al., 2014). They close with increasing level of the ATP/ADP ratio. The question arises, however, how the same KATP channels can regulate two opposite-acting hormones, glucagon at low and insulin at high glucose levels. The mechanism in beta cells predicts that the high glucose is related to the ATP production, closure of KATP channels, cell depolarization, and Ca2+ -induced insulin release from beta cells (Henquin, 20 0 0). In alpha cells, however, the KATP channels are already virtually closed at low glucose. Experimental data show that the net KATP -channel conductance at 1 mM glucose is around 50 pS, which is only around 1 % of that in beta cells (3-9 nS) (Göpel et al., 20 0 0; Zhang et al., 2013, 20 08). Therefore, in hypoglycemia (e.g., 1 mM of glucose) alpha cells are electrically active and secrete glucagon. When glucose is rising, and the ATP concentration is increasing (Li et al., 2015), the open probability of KATP channels is further decreasing, causing membrane depolarization, closing the voltage-dependent Na+ channels, and decreasing the amplitude of action potential firing. This, in turn, reduces the amplitude of P/Q-type Ca2+ -currents and glucagon secretion (Zhang et al., 2013). The pancreatic beta-cell electrical activity and insulin secretion were extensively studied (for a recent review see (Rorsman and Ashcroft, 2018)). Several mathematical models for glucose sensing and glucose-dependent insulin secretion in beta cells have been developed (Bertuzzi et al., 2007; Palumbo et al., 2013; Pedersen, 2009). Closing of the KATP channels is in these models generally accepted as the key switching-on process for insulin secretion (Rorsman et al., 2014). Although the mechanisms of metabolite sensing and the processes leading to glucagon secretion in alpha cells are far less known, several models have also been developed

V. Grubelnik, J. Zmazek and R. Markovicˇ et al. / Journal of Theoretical Biology 493 (2020) 110213

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Fig. 1. Schematic presentation of the model with the main mechanisms included. Glucose and FFA represent the two basic input metabolites for both alpha and beta cells. The main pathways from the metabolite uptake to hormone secretion are indicated. In particular, the aerobic and anaerobic metabolic pathways are emphasized.

for electrical activity and signaling pathways leading to glucagon secretion in alpha cells (Diderichsen and Göpel, 2006; Montefusco and Pedersen, 2015; Pedersen et al., 2017; Watts and Sherman, 2014). Here, we extend the previous models for both alpha and beta cells. We combine the existing mechanisms for glucagon and insulin secretion with the metabolic processes in the pancreatic cells. The metabolism of glucose and FFA are explicitly modeled by integrating the processes of glycolysis and the mitochondrial oxidation of glucose products and FFA. We show that glucagon secretion at low glucose is possible due to FFA oxidation in alpha cells, and at higher glucose, the glucagon secretion is switched off due to the contribution of glycolysis. In beta cells, on the other hand, the oxidative fate of glucose in mitochondria is responsible for the energy switch of insulin secretion. 2. Mathematical model We conducted a computational model incorporating the key mechanisms from oxygen and metabolite uptake, particularly glucose and FFA, to hormone secretion in pancreatic alpha and beta cells. The schematic presentation of the model is given in Fig. 1. The first part of the model encompasses glycolysis, which is a metabolic pathway consisting of ten enzyme-catalyzed reactions converting glucose to pyruvate. During this step, ATP and NADH molecules are produced anaerobically. Pyruvate can be transformed into lactate and leave the cell or can enter mitochondria for further oxidation. Mitochondria are the second major part of the model, and they are crucial for the generation of energy by oxidizing NADH and FADH2 molecules to produce ATP by the electron transfer chain. Two important metabolic pathways by which reducing agents are produced take place in the mitochondrial matrix. Pyruvate produced by glycolysis is transported into mitochondria and can be converted to acetyl-CoA to enter TCA cycle, a cataplerotic pathway producing GTP, NADH and FADH2 molecules. Alternatively, pyruvate can be carboxylated by the pyruvate car-

boxylase (PC) to enter anaplerotic reactions. The second important metabolic pathway, taking place in the mitochondrial matrix, is FFA beta-oxidation, by which NADH, FADH2 , and acetyl-CoA molecules are produced. The latter enter the TCA cycle for further oxidation. ATP produced by the cellular metabolism is then hydrolyzed by various processes such as protein synthesis, Na+ /K+ ATP-ases, Ca2+ ATP-ases, etc. (Buttgereit and Brand, 1995; Lane, 2010). This interaction between cytosolic ATP concentration and the activity of the ATP-consuming processes are also included in the model. The metabolism in pancreatic cells have been included in previous mathematical models, and both the importance of glycolysis and the mitochondrial respiration has been emphasized (Bertram et al., 2006). The metabolism in alpha and beta cells exhibits periodic oscillations, most likely being inherent to reaction dynamics or resulting from negative feedback of increased calcium concentration (Bertram et al., 2007; Kennedy et al., 2002). By rather focusing on the average values of the metabolic fluxes, these lowfrequency oscillations are omitted in the model. A similar approach has been applied in previous studies, e.g. a glucose flux model for beta-HC9 cells incorporates experimental data for glucose uptake, phosphorylation, usage, oxidation, oxygen consumption, and lactate production (Liang et al., 1996). In a model for oxidative phosphorylation (Wilson et al., 2017), glucose-dependent flux functions are integrated based on the experimental data (Doliba et al., 2012). In the model, the metabolic-electrical coupling via KATP -channel activity, which is modulated by the ATP/ADP ratio, plays an important role. Both alpha and beta cells are assumed to possess intrinsic mechanisms of hormone secretion by which the glucosedependent ATP/ADP ratio regulates the membrane potential and the action potential spike characteristics. The membrane potential regulates hormone exocytosis via increased Ca2+ concentration due to the activity of the voltage-dependent Ca2+ channels (VDCC). For alpha cell, we combine the proposed mathematical model of aerobic and anaerobic ATP production with another model, which simulates alpha cell electrical/Ca2+ activity and

V. Grubelnik, J. Zmazek and R. Markovicˇ et al. / Journal of Theoretical Biology 493 (2020) 110213

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1 NADH molecule gets oxidized to NAD+ . The remaining NADH molecules enter mitochondria via glycerol-phosphate and malateaspartate shuttles (MacDonald, 1995; MacDonald et al., 1996).

Table 1 Parameter values used in the model for pancreatic alpha and beta cells (Eqs. 1-18).

Jmax Km pL pTCA JO2, 0 kATPase Km, ATPase Atot g¯ KAT P gS, 1/2 gKATP ,ms ns RGS0 , RIS0

alpha cell

beta cell

7.2 μM/s 2.5 mM 0.9 1 16 μM/s 135 μM/s 2000 μM 4000 μM 54 nS 0.22 nS 0.27 nS 10 0.03

8 μM/s 3.8 mM 0.05 0.4 6 μM/s 180 μM/s 2000 μM 4000 μM 70 nS 0.36 nS 0 nS 4 0.03

2.2. Mitochondrial metabolism

corresponding glucagon secretion (Montefusco and Pedersen, 2015). For beta cells, however, we developed an analogous relationship based on experimentally measured dependencies between glucose, relative insulin secretion rates, and the KATP conductance (Buchwald and Cechin, 2013; Göpel et al., 1999; Pedersen et al., 2011). A more detailed description of particular model components is given in the continuation. The subparagraphs are organized according to the key processes being integrated into the model. The quantitative evaluation of the processes, in the form of mathematical equations, are given together with the explanation of particular terms and factors. The parameter values used in the equations are chosen accordingly to known experimental values, measured mostly in rodent alpha and beta cells. The model parameters are listed in Table 1. 2.1. Glycolysis Glucose is initially transported into intracellular space and phosphorylated to G6P during the first priming reaction of glycolysis. The processes are considered to be the rate-limiting step of the glycolysis (German, 1993; Gloyn et al., 2005; Heimberg et al., 1996) and therefore determine the glycolytic flux. Glucose uptake and the phosphorylation are modeled by the Michaelis-Menten kinetics as follows:

JG6P = Jmax

2 Km

G2 . + G2

(1)

Parameters Jmax and Km are chosen based on the experimental data for glucose concentrations G at 1 mM and 10 mM (Schuit et al., 1997), as indicated in Fig. 2A. Values for glucokinase activity and glycolytic flux are comparable in alpha and beta cells (Heimberg et al., 1996, 1995). Taking into account the preparatory and the payoff phase, glycolysis yields a net total of 2 ATP and 2 NADH molecules, which is captured by the following equations:

JATP,Gly = 2JG6P ,

(2)

JNADH,Gly = 2JG6P ,

(3)

Mitochondria, incorporating aerobic metabolism in the vast majority of eukaryotic cells, is crucial for making the cell’s metabolism highly energy-efficient at the expense of the necessity for the transport of oxygen and carbon dioxide. In a series of papers in the late 1990s, Magnus and Keizer developed mathematical models of oxidative phosphorylation (Magnus and Keizer, 1998a, 1998b, 1997) that has been further modified by Bertram et al. (2006). By the mitochondrial metabolism, as much as 32 ATP molecules can be produced from 1 molecule of glucose. Additionally, mitochondria compartmentalize the FFA beta-oxidation reactions, separating them from FFA synthesis, taking place in the cytosol. Firstly, the model for the oxidative fate of glucose is presented. It is comprised of the pyruvate oxidation by the TCA cycle and the oxidation of NADH molecules produced in glycolysis and TCA cycle. Secondly, the oxygen consumption rate is modeled in both cell types. Lastly, the model describes the oxidative fate of FFA. 2.2.1. Oxidative fate of glucose A fraction of pyruvate produced by glycolysis enters mitochondria for further oxidation. The model separates ATP production depending on where NADH molecules were produced: ATP from glycolysis-produced NADH (JATP,NADH,Gly ) and ATP from TCA-cycleproduced NADH (JATP,NADH,pyr ):

JAT P,GO = JAT P,NADH,Gly + JAT P,NADH,pyr .

(5)

While beta cells express very low quantities of LDH (Sekine et al., 1994), this is not the case for alpha cells, which express a 4-fold higher ratio of LDH to FAD-linked GPDH (Schuit et al., 1997). A fraction of pyruvate is therefore lost due to conversion of pyruvate to lactate, depending on the cell type. We model this effect by introducing a parameter pL and taking values pL = 0.05 for beta cell and pL = 0.9 for alpha cell. Transport of pyruvate into mitochondria, oxidation of pyruvate to acetyl-CoA and production of reducing equivalents in TCA cycle are modeled by Eq. (6), in which we make a simplifying assumption that 4 NADH, 1 FADH2 and 1 GTP molecules produced by the TCA cycle are equal to 5 NADH molecules from the energy standpoint. The simplification is based on the assumption that oxidation of 1 FADH2 in electron transport chain yields 1.5 ATP, while oxidation of 1 NADH yields 2.5 ATP (P/O ratio). A portion of pyruvate is also lost due to anaplerotic reactions. This effect is taken into account with the parameter pTCA , which differs between alpha and beta cell. In alpha cells, anaplerosis is not detectable (Schuit et al., 1997), which is modeled by pT CA = 1. Previous studies suggest that anaplerosis in beta cells is relatively important, but there is a large discrepancy in the experimental data (Doliba et al., 2012; Liang et al., 1996; Schuit et al., 1997). We approximate the anaplerotic flux by considering pT CA = 0.4 as follows:

JNADH,pyr = 5 pTCA (1 − pL )Jpyr .

(6)

where JATP, Gly is glycolysis net ATP production flux and JNADH, Gly is the flux of NADH that later contributes to glycolytic ATP production. Glycolysis-produced ATP is also a measure of anaerobic ATP production. The final products of glycolysis are two molecules of pyruvate, which is modeled as:

Taking into account the P/O ratio of 2.5, which is typical for the NADH oxidation in the electron transport chain, the rate of ATP production from pyruvate-produced NADH molecules is given by:

J pyr = 2JG6P .

Glycerol–phosphate shuttle and mitochondrial glycerolphosphate dehydrogenase (mGPDH) activities are very high in beta-cells relative to other mammalian cells (MacDonald, 1990). The lack of NAD+ -linked glycerol phosphate dehydrogenase does

(4)

Depending on the amount of the LDH expressed in the cell, a portion of pyruvate is reduced to lactate in the reaction by which

JATP,NADH,pyr = 2.5JNADH,pyr .

(7)

V. Grubelnik, J. Zmazek and R. Markovicˇ et al. / Journal of Theoretical Biology 493 (2020) 110213

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Fig. 2. Evaluation of the model: comparing the model dynamics with experimental data. A) Glycolytic flux JG6P in alpha (red line) and beta (black line) cell with corresponding experimental measurements reported by Schuit et al. (Schuit et al., 1997) (red and black dots with error bars). B) Alpha (red line) and beta (black line) cell’s oxygen consumption rate as predicted by our model and black squares signify experimental data for beta cells, as measured by Doliba et al. (Doliba et al., 2012). In the lower panel, the red line shows the increase of ATP for about 10 % when glucose concentration increases from 1 mM to 5 mM. Red circles denote the experimentally measured values by Li et al. (2015).

not abolish normal beta-cell function in mice (MacDonald and Marshall, 20 0 0), but by blocking both NADH-shuttles, the GSIS is abrogated (Eto et al., 1999). This indicates that for the normal beta-cell function, both NADH shuttles are necessary. However, the activity of the malate-aspartate shuttle, which transfers NADH molecules directly, is NAD/NADH ratio-dependent (Dukes et al., 1994) and might be therefore less important in hyperglycemic conditions. While the malate-aspartate shuttle transfers NADH molecules directly, glycerol-phosphate shuttle transports electrons from NADH (by regenerating NAD+ ) to glycerol-3-phosphate dehydrogenase 2, reducing enzyme-bound FAD to FADH2 . Therefore, the P/O ratio for the glycerol-phosphate shuttle is only 1.5, in contrast to the malate-aspartate shuttle, for which the P/O ratio is 2.5. For simplicity of the model, we omit the NAD+ /NADH dependency and assume only glycerol-phosphate shuttle activity. Accordingly, the rate of ATP production from the glycolysis-produced NADH molecules is given by:





JAT P,NADH,Gly = 1.5 JNADH,Gly − pL J pyr .

(8)

Molecular oxygen is the final acceptor of electrons in the electron transport chain and is required for the aerobic respiration. The amount of oxygen consumption due to the electron transport chain activity from glucose oxidation is modeled by:

JO2,G =

 1 JNADH,pyr + JNADH,Gly − pL Jpyr . 2

(9)

The percentage of glucose that is oxidized can be estimated by the following expression:

PG =

1 1 1 (1 − pL ) pT CA + (1 − pL ) = (1 − pL )(1 + pT CA ), 2 2 2

(10)

where pL and pTCA are the same as defined in Eq. (6). The model yields pG = 10 % for alpha cell and pG = 67 % for beta cell, which is in broad agreement with experimental measurements (Gorus et al., 1984; Quesada et al., 2006). 2.2.2. Oxygen consumption Oxygen consumption in beta cells increases with the acceleration of glycolysis, while it decreases in most other cell types, in

accordance to the so-called Crabtree effect (Schuit et al., 1997). In beta cells, the oxygen consumption JO2 is modelled by Eq. (11a) following the model by Wilson et al. (2017), which fits the experimental data from (Doliba et al., 2012):

JO2 = JO2,1

JG6P nO2 nO2 Km,O2 + JG6P nO2

+ JO2,0 ,

(11a)

where nO2 Km,O = 12 mM, nO2 = 1.5, JO2,0 = 6 2

μM s

, and JO2,1 = 50

μM s

.

Because of lack of experimental data for alpha cells, oxygen consumption is modeled according to the following observations: the Crabtree effect, observed in alpha cells (Schuit et al., 1997), indicates that oxygen consumption decreases with increasing glucose levels. At low glucose concentrations, alpha cells use more oxygen than beta cells. This follows from the fact that in hypoglycemic conditions, alpha cells are electrically much more active than beta cells and secrete more hormone granules. Based on the experimental data showing that ATP concentration increases by approximately 10 % when glucose rises from 1 mM to 5 mM (Li et al., 2015), the rate of oxygen consumption in the alpha cell is modeled as the sum of two contributions:

JO2 = kO2,G JG6P + JO2,0 ,

(11b)

where

kO2,G = −0.2 and JO2,0 = 16

μM s

.

The parameter values for kO2, G and JO2, 0 provide the best fit of the Eq. (11b) to the available experimental data for oxygen consumption in alpha cells (Fig. 2A, red dots). In hypoxic conditions (1 mM glucose), the experimental data indicate larger oxygen consumption in alpha than in beta cells, which corresponds to the higher FFA oxidation in alpha cells, as proposed in our model. It is also known that a greater amount of O2 is required per mol of ATP obtained from the oxidation of FFA compared to the oxidation of glucose (Diao et al., 2008; Jaswal et al., 2009).

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In Fig. 2, we explicitly show how the proposed model kinetics for alpha and beta cells match the experimental data from the literature. The magnitudes of the glycolytic fluxes JG6P at low and high glucose concentrations fit well the data from Schuit et al. (1997), as indicated by red and black dots in Fig. 2A. Moreover, the oxygen consumption rate for the beta cell, which is presented by the black line in Fig. 2B matches the experimental values (black squares) from Doliba et al. for three different glucose concentrations (Doliba et al., 2012). The ATP production in alpha cells (red line in Fig. 2B) resembles the experimental data reported by Li et al. (2015). 2.2.3. Oxidation of free fatty acids In addition to the ATP production from glucose oxidation, ATP is also produced by the FFA beta-oxidation pathway. In both alpha and beta cells, FFA have been demonstrated to increase insulin and glucagon secretion (Itoh et al., 2003; Olofsson et al., 2004). While in beta cells FFA may regulate insulin secretion via GPR40 signaling pathway with beta-oxidation being less important (Kristinsson et al., 2017), beta-oxidation of FFA in alpha cells is CPT1a dependent and the pharmacological blockade of CPT1 reduces glucagon secretion by approximately 40 % (Briant et al., 2018a). Beta-oxidation-produced NADH is approximated by the difference between total oxygen consumption and oxygen consumption due to glucose oxidation, which is given by:

JNADH,FFA = 2(JO2 − JO2,G ).

(12)

Accordingly, the rate of ATP production from the betaoxidation-produced NADH is modeled by:

JATP,FFAO = 2.5JNADH,FFA ,

(13)

again, assuming the P/O ratio of 2.5. 2.3. Conductance of KATP channels The cytosolic ATP concentration is determined by a balance between the ATP production and consumption. ATP is produced by glycolysis (anaerobic ATP production) and by the electron transport chain, which oxidizes reducing equivalents from glycolysis, pyruvate oxidation and FFA oxidation (aerobic ATP production). The rate of the ATP production is given by:

JAT P = JAT P,Gly + JAT P,GO + JAT P,F F AO .

(14)

It is generally assumed that ATP hydrolysis increases with cell’s energy state and the kinetics is often modelled accordingly to Michaelis-Menten kinetics. A similar approach has been used previously (e.g. Bertram and Sherman, 2004). The rate of the ATP hydrolysis is described by:

JAT Pase = kAT Pase

AT P . Km,AT Pase + AT P

(15)

The concentration of ATP in the steady-state approximation, when ATP production rate, JATP , is in equilibrium with the rate of ATP hydrolysis, JATPase , is given by:

AT P =

Km,AT Pase JAT P kAT Pase − JAT P

(16)

The KATP channels represent a coupling step between cell’s metabolic and electric activities. ATP directly inhibits KATP channel by binding to Kir6.2 subunits, while ATP and ADP activate the channel through interaction with the NBFs of SUR (Nicholls, 2016). Therefore, KATP -channel activity is ATP/ADP dependent – higher ATP/ADP ratio reduces KATP -channel activity. The KATP channel conductance is modelled as proposed by Magnus and Keizer (1998a,b):



gKATP = gKATP

2 MgADP − 17μM MgADP − 2

0.08 1 +



1+

17μM





+ 0.89

1+

ADP 3− 26μM

 MgADP− 2 17μM

4−

P + AT 1 μM

 ,

(17)

where

MgADP − = 0.165 ADP, ADP 3− = 0.135 ADP, ATP 4− = 0.05 ATP. Concerning the value of g¯ KAT P in Eq. (17), we take the originally proposed value for beta cells, gKATP = 70 nS (Magnus and Keizer, 1998a), whereas for alpha cells this value is set to 54 nS, resembling the experimental data showing that the whole-cell KATP channel conductance is smaller in alpha than in beta cells (Barg et al., 20 0 0). 2.4. Hormone secretion Lastly, we incorporate the mechanisms for the secretion of glucagon in alpha cells and the secretion of insulin in beta cells. For both cell types, a large number of studies has shown that KATP channels play a pivotal role in the process of regulating hormone secretion (Bratic and Trifunovic, 2010; MacDonald et al., 2007; Östenson et al., 1980; Quesada et al., 2006; Zhang et al., 2013). Following these findings, we link exocytosis with the conductance of KATP channels, which have been shown to be identical in both cell types (Diao et al., 2008; Gromada et al., 2004; Olsen et al., 2005; Östenson et al., 1980; Quesada et al., 2006; Rorsman et al., 2014; Zhang et al., 2013). To this purpose, we implement previous studies, which describe glucagon and insulin secretion. For glucagon secretion, we use the mathematical model developed by Montefusco and Pedersen (2015), which enables detailed simulation of Ca2+ dynamics and the exocytosis downstream of the electrical activity. By implementing this model, we have established the relation between the gKAT P and the relative glucagon secretion (Fig. 3A, black line). On the other hand, for beta cells, the relation between the gKAT P and the relative insulin secretion was determined by using the model developed by Pedersen et al. (2005), see Fig. 3B (black line). For the sake of completeness, the model equations taken from the previous models (Montefusco and Pedersen, 2015; Pedersen et al., 2005) with the complete list of model parameters used in our calculations are provided in the Supplement. In order to simplify the coupling between the models for insulin and glucagon secretion (Pedersen et al., 2005; Montefusco and Pedersen, 2015) with the bioenergetic model (Eqs. (1)–(17)), we introduce a phenomenological relationship between the KATP channel conductance and the hormone secretion by fitting the original model predictions (Fig. 3, black lines) with the following equations (Fig. 3, red lines):



RGS = (1 − RGS0 )

fRGS gKATP



fRGS gKATP ,ms



RIS = (1 − RIS0 )



fRIS gKATP



 + RGS0 ,

(18a)



fRIS gKATP ,ms

 + RIS0 ,

(18b)

where





fRGS gKATP =





fRIS gKATP =

gnKsATP gnS,s1/2 + gnKsATP gnS,s 1/2 gnS,s1/2 + gnKsATP

,

(18c)

.

(18d)

The parameter gKATP ,ms in Eqs. (18a,b) represents the KATP channel conductance corresponding to the maximal hormone secretion. For alpha cells, the value of gKATP ,ms is set to 0.27 nS, as experimentally measured by Zhang et al. (2013) and used in the model by Edlund et al. (2017), whereas for beta cells gKATP ,ms → 0. The parameter values for Eqs. (1-18) are listed in Table 1.

V. Grubelnik, J. Zmazek and R. Markovicˇ et al. / Journal of Theoretical Biology 493 (2020) 110213

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Fig. 3. Relative hormone secretion (black and red lines) with the corresponding electrical activity (blue lines) of alpha (A) and beta (B) cells in dependence on the KATP channel conductance, gKAT P . The black lines correspond to the results of the original models for alpha (Montefusco and Pedersen, 2015) and beta (Pedersen et al., 2005) cells, respectively. The red lines were calculated with Eqs. (18a-d).

3. Results With the computational model (Eqs. 1-18) that links the metabolism of pancreatic alpha and beta cells with the exocytotic machinery, we study the mechanisms of glucose-dependent glucagon and insulin secretion. To evaluate the contributions of glucose products and FFA to ATP production, we analyze the fluxes of metabolites throughout the metabolic pathways. Finally, we assess the cell’s energy state, which is principally determined by the input and the output of the high-energy ATP molecules. The energy fluxes in alpha and beta cells thus determine the energy-driven switches for glucagon and insulin secretion. Understanding the interplay between these fluxes will help us to understand the secretion patterns, i.e. that glucagon is secreted at low glucose levels and switched off above a certain level of glucose concentration, whereas on the other hand, the secretion of insulin is switched on at approximately the same threshold concentration, as observed experimentally (Gylfe and Gilon, 2014; Rorsman et al., 2014, 2008). 3.1. Energy fluxes in alpha and beta cell Fluxes of ATP production as a function of glucose concentration are shown in Fig. 4 for both cell types. We distinguish between: i) the complete flux of ATP production (JATP ; black line), which determines the cell energy state, ii) the flux of ATP production from aerobic metabolism of glucose (JATP,GO ; red line), and iii) the flux of ATP production from aerobic FFA oxidation (JATP,FFAO ; green line). Additionally, the sum of both fluxes of aerobic ATP production from glucose (JATP,GO +JATP,FFAO ; magenta line) is presented. The sum reflects the cell’s oxygen consumption rate as well as the rate of anaerobic (glycolytic) ATP production (JATP,Gly ; blue line). Results in Fig. 4A show that the complete flux of ATP production, JATP , which is derived from glycolysis (anaerobic ATP production) and oxidative phosphorylation of NADH from glucose and FFA oxidation (aerobic ATP production), exhibits poor glucose dependency in alpha cells. The main cause for this weak glucosedependency of JATP , despite comparable values of glycolytic fluxes and glycolysis-derived ATP, is a large output of high-energy lactate derived from pyruvate produced by glycolysis (see Eqs. (6)–

(8)). Our computational results indicate that FFA oxidation is reduced with increasing glucose concentration, which is consistent with the importance of the FFA during hypoglycemia. Total aerobic ATP production (magenta line) in the alpha cell slightly declines with the increasing glucose concentration, which is consistent with the decreasing oxygen consumption (see Fig. 2B) modeled according to the Crabtree effect (Schuit et al., 1997). However, in total the ATP production, JATP still increases with increasing levels of glucose due to the contribution of the anaerobic flux of ATP production (JATP,Gly ). Results shown in Fig. 4B reveal that in contrast to the alpha cell, the complete flux of ATP production JATP of beta cell exhibits significantly greater glucose-dependency. Lactate output in the beta cell is minimal and high amount of pyruvate can be further oxidized to generate more ATP. The degree of glucose-dependency is, however, slightly reduced due to the higher anaplerotic output in the beta cell and also due to the FFA oxidation which, similarly as in alpha cell, declines at higher glucose concentrations. These results are in agreement with the well-known regulation of Randle cycle (glucose fatty-acid cycle), a mechanism involving the selection of preferred fuel source for respiration (Guo, 2015; Hue and Taegtmeyer, 2009). The glucose-dependent ATP production is therefore mainly due to the JATP,GO (Fig 4B, red line and Eq. 5), which includes the production of ATP from the glycolysis-produced NADH and TCA cycle-produced NADH. The increase of aerobic ATP production with increasing glucose concentration is also consistent with the increase in oxygen consumption (Fig. 2B). 3.2. Energy-driven switch in alpha and beta cells Hormone secretion from alpha and beta cells is crucial for homeostatic blood glucose regulation. Under hypoglycemia, alpha cells need to secrete glucagon to increase blood glucose levels, and under hyperglycemia, beta cells secrete insulin to lower the blood glucose. This switch-like behavior is energy-driven and requires sufficient ATP production to stop/start the exocytosis. Based on the comparison between the complete ATP production, the aerobic ATP production, and the anaerobic ATP production, presented in Fig. 4, the significance of the aerobic and anaerobic compo-

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Fig. 4. Fluxes of ATP production as a function of glucose concentration in alpha (A) and beta (B) cell. Total ATP production JATP exhibits significantly larger glucosedependency in beta than in alpha cell (black line). Alpha cells also oxidize more FFA during hypoglycemic conditions and the FFA consumption declines with increasing glucose concentration (JATP,FFAO ; green line). JATP,Gly are comparable because of similar glycolytic fluxes in both cell types (blue line). The beta cell uses significantly less pyruvate and glycolysis-produced NADH for the production of lactate in contrast to the alpha cell. Consequently, the beta cell produces more NADH (JATP,GO ; red line).

nent in each cell can be assessed. In particular, we are interested in the aerobic/anaerobic energy contribution at the threshold glucose level, 6 mM, where the switching point (SP) is defined. This point highlights a stimulatory glucose concentration level at which a cell changes from a hormone-secreting state (ON state) to a nonsecreting state (OFF state), or vice versa. In alpha cells, aerobic ATP production alone is not enough to drive the cell over the switching point (see Fig. 5A). At low levels of glucose, aerobic ATP production is saturated and is hence unable to produce the necessary energy flow to exceed the energy level needed to overpass the switching point. By additionally increasing stimulatory glucose levels and increasing anaerobic metabolic fluxes, the net energy is elevated to the level required for switching from one regime to another, the so-called “anaerobic switch”. This anaerobic switch is efficient even though the increasing anaerobic energy production decreases the aerobic production in mitochondria. In beta cells (see Fig 5B), on the other hand, the mitochondrial aerobic energy production governs the operation regime of the beta cell. Increasing glucose levels predominately leads to an increase in mitochondrial glucose oxidation, since mitochondria are not saturated at low glucose levels and can be activated by the higher glucose concentrations. Therefore, the energy-driven switch in beta cells could be characterized as “aerobic switch”. Finally, the interplay between the glucose stimulation, the KATP channel conductance, and the resulting hormone secretion can be presented. The results are shown in Fig. 6. In both alpha and beta cells, the KATP -channel conductance gKAT P , is decreasing with the elevated glucose concentration (Fig. 6A), as expected (Rorsman et al., 2014). The corresponding relative secretion rates of glucagon and insulin are also well in agreement with experimental observations (Buchwald and Cechin, 2013; Walker et al., 2011; Zhang et al., 2013). The results show that the energy, in the form of ATP, is crucial for modifying the conductance of the KATP channels, and consequently, the hormone secretion is switched ON/OFF in terms of an efficient energy-driven switch.

To further analyze the efficiency of these metabolic switches in alpha and beta cells, we investigate how the sensitivity of the switches depend on the energy production by anaerobic/aerobic metabolism in the cells. The results concerning the sensitivity of both switches might importantly contribute to better understanding of the pathologies developed by impaired bioenergetics in alpha and beta cells. In Fig. 6B the effect of glycolysis on glucagon and insulin secretion is presented. The results are obtained by varying the amount of glycolysis-derived ATP molecules. In the model, the simulation of relative inhibition of glycolysis, and hence its contribution to the total ATP production in the cell, is introduced by multiplying the Eq. (2) with a new parameter pGly . Following the model, the decreased ATP/ADP ratio, due to the inhibition of glycolysis, influences KATP -channel conductance and finally the hormone secretion. The results show that the glycolysis is crucial for the switch operation in alpha cells, where a decrease by nearly 40 % (pGly = 0.6) completely abolishes the switch operation. The reason for this is that the KATP channels in alpha cells cannot close sufficiently and glucagon secretion is not diminished. Conversely, changing the glycolysis-derived ATP flux by the same amount in the beta cell does not significantly influence insulin secretion. This again advocates the aerobic characterization of the switch in beta cells. 4. Discussion The presented mathematical model links the mechanisms of glucagon and insulin secretion with the metabolic processes in pancreatic alpha and beta cells. The metabolic processes, particularly the metabolism of glucose and FFA, are explicitly modeled by integrating the key mechanisms of glycolysis and mitochondrial oxidation of glucose products and FFA. This enables to trace the metabolic fluxes and to estimate the ATP levels in the cells. The model predictions show that in alpha cells, the energy flux required for glucagon secretion at low glucose is mainly provided by

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Fig. 5. Energy-driven switch in alpha (A) and beta (B) cells. In alpha cells, this switch is driven by anaerobic ATP production, whereas in beta cells the key contributor to the switch-like behavior is the aerobic ATP production. The red horizontal line represents the energy threshold level required for the switch operation between ON (secreting) and OFF (non-secreting) state. The corresponding glucose concertation (6 mM) is denoted with a black vertical dotted line. The crossing point is referred to as the switching point (SP).

Fig. 6. Relative insulin secretion (RIS) and relative glucagon secretion (RGS) as a function of glucose concentration. A) The level of hormone secretion and the conductance of KATP channels (gKATP ) for both cell types. B) Effect of relative inhibition of glycolysis on glucagon and insulin secretion.

FFA oxidation. At higher glucose concentrations, the glucagon secretion is switched off due to the contribution of glycolysis. The rate of glycolysis is very similar in alpha and beta cells; however, the fate of glucose is very different in the two cells. Whereas in alpha cells the anaerobic metabolism of glucose is crucial for turning off the glucagon secretion, in beta cells the oxidative fate of glucose in mitochondria is the key component of the energy switch for insulin secretion in healthy conditions. This result agrees

with previous studies indicating that the mitochondrial activation in beta cells is the key event in the process of glucose-mediated insulin secretion (Kennedy et al., 1998; Maechler and Wollheim, 2001; Quesada et al., 2006). Recent findings also show that the pathology in diabetic mouse is closely related to the metabolic changes in beta cells that markedly reduce glucose metabolism in mitochondria and, consequently, the ATP synthesis (Haythorne et al., 2019). This breaks down the old paradigm and suggests that

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the primary cause of the insufficient insulin secretion in T2DM is impaired metabolism-secretion coupling rather than beta-cell loss (Cantley and Ashcroft, 2015; Haythorne et al., 2019; Rahier et al., 2008; Weir and Bonner-Weir, 2013). The model prediction, indicating that the anaerobic glycolysis essentially contributes to the regulation of glucagon secretion in alpha cells, agrees with previous experimental observations showing that an inhibitor of glycolysis, mannoheptulose, suppresses glucose-induced glucagon secretion (Olsen et al., 2005). The energy production by the glycolysis is apparently crucial for the normal functioning of alpha cells. Moreover, Olsen et al. (Olsen et al., 2005) have shown that, although mannoheptulose suppresses glucose-induced glucagon secretion, it does not influence the basal glucagon secretion in isolated alpha cells. In our model, this fact can be explained by the process of FFA oxidation in mitochondria that provides the required quantities of energy for keeping the KATP channels almost closed and secreting the glucagon. Therefore, the efficient functioning of mitochondria, using FFA as the main fuel in hypoglycemic conditions, is crucial for the basal glucagon secretion (Briant et al., 2018a). In hyperglycemic conditions above the threshold level, however, our model predictions show that glucose metabolism is the main contributor to the energetic switch turning off the glucagon secretion. Considering the ATP-producing glucose metabolism incorporates the anaerobic and the aerobic part of the glucose fate. In general, it is known that glucose is mostly oxidized in beta, and to much less extent in alpha cells. The glucose oxidative capacity of alpha cells is estimated to be only 1/3 to 1/6 of that in beta cells (Schuit et al., 1997). Our model fits these proportions quite well; however, it is not completely clear to what extent the anaerobic vs. aerobic fate of glucose contributes to the glucagon secretion in alpha cells. Experiments have been provided with oligomycin, a blocker of the mitochondrial ATP synthase, in which the glucagon secretion has been measured in dependence on glucose. The exclusion of the oxidative fate of glucose in mitochondria showed a clear impact on glucagon secretion, even converting the inhibitory effect of glucose on glucagon secretion into stimulation, which clearly shows a regulatory role of non-mitochondrial metabolism in alpha cells (Zhang et al., 2013). The impact of this non-mitochondrial metabolism on glucagon secretion is quantified in our model, showing that its contribution to the ATP production is indispensable for the regulation of glucagon secretion at the threshold glucose level, switching off the glucagon secretion, and hence representing the so-called anaerobic switch. The intrinsic hormone-secretion mechanism in alpha and beta cells, as presented in our model, is well supported experimentally; in particular, the experiments with diazoxide, an activator of KATP channels (MacDonald et al., 2007; Zhang et al., 2013), show the important role of energy-driven regulation of hormone secretion. Moreover, the experimental measurements of glucagon secretion at different glucose concentrations have shown that the relative glucagon secretion is reduced by approx. 60 % when glucose was changed from 1 mM to 7 mM and 8.3 mM in mouse and rat, respectively (MacDonald et al., 2007). Similar results have been obtained by Zhang et al. (2013) revealing a 55% reduction of glucagon secretion when glucose was increased from 1 mM to 6 mM. They have also measured the corresponding KATP -channel conductance, gKAT P , and the data show that the value of gKATP takes a value of about 0.27 nS at 1 mM glucose and reduces down to 0.20 nS at 6 mM glucose. All these experimental data are well reproduced by our mathematical model, see Fig. 6. The paracrine regulation of the pancreatic hormone secretion also plays a role (Watts et al., 2016). In particular, for alpha cells, the influence of paracrine factors is still obscure. Experimental measurements of glucagon secretion from pure populations of flow-sorted alpha cells indicate that only Zn2+ could potentially

have an inhibitory effect on glucagon secretion in mouse alpha cells (Le Marchand and Piston, 2010). The experiments didn’t show any other effects among all the other candidate paracrine inhibitors used: insulin, GABA, and somatostatin. Conversely, other experiments on genetically modified mouse have shown that glucose can inhibit glucagon release independently of Zn2+ , and also with diminishing effects of KATP channels and somatostatin (Cheng-Xue et al., 2013; Gylfe, 2013). Recent experiments (Briant et al., 2016, 2018b) show that an interplay between alpha, beta, and delta cells is important for glucagon secretion. These aspects, however, should be considered when the model is used for studying islet dynamics, especially at higher glucose concentrations. Recent experiments using glucose and KATP blockers on perfused islets show that the glucagonostatic effect of glucose is mainly independent for low glucose (0-7 mM) but starts to involve somatostatin for higher glucose concentrations (15–30 mM) (Lai et al., 2018). Therefore, any further applications of our model for higher glucose levels would need to consider the paracrine effects of somatostatin. Further extensions of our model could be made in a more detailed description, in particular in the specification of the ATPases. The Na+ /K+ -pumping ATPase is a major energy consumer in most types of cells, and they have an important impact on the electrophysiological properties of alpha and beta cells, e.g., a reduced Na+ /K+ pumping results in a more positive equilibrium potential for K+ (Gylfe, 2016). Briant et al. (2018a) have explicitly modeled the activity of Na+ /K+ pump; so, this part could be directly implemented as a plug-in in further models. The ATPases pumping Ca2+ into the intracellular stores also play an important role and could be explicitly included into the model (Gromada et al., 2007; Liu et al., 2004; Vieira et al., 2007). A more detailed modeling of energetic processes in alpha and beta cells would also be valuable according to the most recent experimental evidence showing that elevations in intracellular Na+ concentration, caused by hyperglycemia in T2DM, leads to acidification as a direct consequence of a lower Na+ gradient across the plasma membrane that cannot drive efficiently the uphill transport of H+ , and via reduction of intramitochondrial pH leads to a lower H+ flux through ATP synthase and to an impaired ATP production (Knudsen et al., 2019). In addition to the main energy fluxes in the cell, other signaling pathways should also be considered. The action of cyclic AMP (cAMP) as a second messenger is probably the most important contribution to the regulation of insulin and glucagon secretion (Elliott et al., 2015; Yu et al., 2019). Although there is less doubt that Ca2+ is required for activation of insulin and glucagon granules, the magnitude of glucagon secretion appears to be mainly controlled by the cAMP-mediated amplification of granule exocytosis (Yu et al., 2019). In the future, these findings would need to be incorporated in more detailed modelling of hormone secretion in alpha and beta cells. Another very important signaling pathway is AMPK, and in addition to the energydriven KATP -channel permeability, the newly discovered AMPKsignal-driven KATP -channel-density regulation should also be considered in future modeling of insulin and glucagon secretion (Han et al., 2018). The mathematical modeling of metabolic and signaling pathways in pancreatic alpha and beta cells is important for better understanding of pancreatic physiology and represents a promising potential for exploring possible future treatments of metabolic diseases, particularly the metabolic syndrome and diabetes. To this aim, further studies will be needed to investigate the complex interplay between the energy-driven, anabolic, and signaling mechanisms. A better understanding of mitochondria as the main bioenergetic, but also biosynthetic and signaling organelles (Chandel, 2015) may represent the first step in completely new clinical treatments of diabetes, oriented into re-establishing the dysregulated

V. Grubelnik, J. Zmazek and R. Markovicˇ et al. / Journal of Theoretical Biology 493 (2020) 110213

cellular mechanisms of the disease. Concerning the mitochondrial processes, we could improve the future treatment of diabetes with the development of new medications stimulating mitogenesis and influencing mitochondrial function more efficiently than currently known agents, e.g., coenzyme Q10 (Hernández-Camacho et al., 2018) or metformin (Cordero et al., 2016). Credit author statement Marko Marhl, Vladimir Grubelnik, and Rene Markovicˇ conceived the idea of the model. Vladimir Grubelnik and Rene Markovicˇ carried out the model calculations. Jan Zmazek participated in evaluating the model parameters. All authors were involved in writing the manuscript; in particular, Marko Marhl wrote the main part of the Introduction and the Discussion, Vladimir Grubelnik and Jan Zmazek the Model and the Results. The Supplement was mainly written by Jan Zmazek and Rene Markovicˇ . Marko Gosak has participated in writing and editing of the manuscript. Funding This work was supported by the Slovenian Research Agency (research core funding, Nos. P1-0055, P3-0396). Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jtbi.2020.110213. References Affourtit, C., Alberts, B., Barlow, J., Carré, J.E., Wynne, A.G., 2018. Control of pancreatic β -cell bioenergetics. Biochem. Soc. Trans. 46, 555–564. doi:10.1042/ BST20170505. Baeyens, L., Lemper, M., Staels, W., De Groef, S., De Leu, N., Heremans, Y., German, M.S., Heimberg, H., 2018. (Re)generating human beta cells: status, pitfalls, and perspectives. Physiol. Rev. 98, 1143–1167. doi:10.1152/physrev.0 0 034.2016. Barg, S., Galvanovskis, J., Gopel, S.O., Rorsman, P., Eliasson, L., 20 0 0. Tight coupling between electrical activity and exocytosis in mouse glucagon-secreting alphacells. Diabetes 49, 1500–1510. doi:10.2337/diabetes.49.9.1500. Bertram, R., Gram Pedersen, M., Luciani, D.S., Sherman, A., 2006. A simplified model for mitochondrial ATP production. J. Theor. Biol. 243, 575–586. doi:10.1016/j.jtbi. 2006.07.019. Bertram, R., Sherman, A., 2004. A calcium-based phantom bursting model for pancreatic islets. Bull. Math. Biol. 66, 1313–1344. doi:10.1016/j.bulm.20 03.12.0 05. Bertram, R., Sherman, A., Satin, L.S., 2007. Metabolic and electrical oscillations: partners in controlling pulsatile insulin secretion. Am. J. Physiol. Metab. 293, E890– E900. doi:10.1152/ajpendo.00359.2007. Bertuzzi, A., Salinari, S., Mingrone, G., 2007. Insulin granule trafficking in β -cells: mathematical model of glucose-induced insulin secretion. Am. J. Physiol. Metab. 293, E396–E409. doi:10.1152/ajpendo.0 0647.20 06. Bouillaud, F., 2009. UCP2, not a physiologically relevant uncoupler but a glucose sparing switch impacting ROS production and glucose sensing. Biochim. Biophys. Acta – Bioenergy 1787, 377–383. doi:10.1016/j.bbabio.2009.01.003. Bratic, I., Trifunovic, A., 2010. Mitochondrial energy metabolism and ageing. Biochim. Biophys. Acta – Bioenergy 1797, 961–967. doi:10.1016/j.bbabio.2010.01. 004. Briant, L.J.B., Dodd, M.S., Chibalina, M.V., Rorsman, N.J.G., Johnson, P.R.V., Carmeliet, P., Rorsman, P., Knudsen, J.G., 2018a. CPT1a-dependent long-chain fatty acid oxidation contributes to maintaining glucagon secretion from pancreatic islets. Cell Rep. 23, 3300–3311. doi:10.1016/j.celrep.2018.05.035. Briant, L.J.B., Reinbothe, T.M., Spiliotis, I., Miranda, C., Rodriguez, B., Rorsman, P., 2018b. δ -cells and β -cells are electrically coupled and regulate α -cell activity via somatostatin. J. Physiol. 596, 197–215. doi:10.1113/JP274581. Briant, L.J.B., Salehi, A., Vergari, E., Zhang, Q., Rorsman, P., 2016. Glucagon secretion from pancreatic α -cells. Ups. J. Med. Sci. 121, 113–119. doi:10.3109/03009734. 2016.1156789. Briant, L.J.B., Zhang, Q., Vergari, E., Kellard, J.A., Rodriguez, B., Ashcroft, F.M., Rorsman, P., 2017. Functional identification of islet cell types by electrophysiological fingerprinting. J. R. Soc. Interface 14, 20160999. doi:10.1098/rsif.2016.0999. Buchwald, P., Cechin, S.R., 2013. Glucose-stimulated insulin secretion in isolated pancreatic islets: Multiphysics FEM model calculations compared to results of perifusion experiments with human islets. J. Biomed. Sci. Eng. 06, 26–35. doi:10.4236/jbise.2013.65A006. Buttgereit, F., Brand, M.D., 1995. A hierarchy of ATP-consuming processes in mammalian cells. Biochem. J. 312, 163–167. doi:10.1042/bj3120163.

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