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In mammals the concentration of blood glucose is kept close to 5 mmol/l. Different cell types in the islet of Langerhans participate in the control of glucose homeostasis. β-cells, the most frequent type in pancreatic islets, are responsible for the synthesis, storage, and release of insulin. Insulin, released with increases in blood glucose promotes glucose uptake into the cells. In response to glucose changes, pancreatic α-, β-, and δ-cells regulate their electrical activity and Ca2+ signals to release glucagon, insulin, and somatostatin, respectively. While all these signaling steps are stimulated in hypoglycemic conditions in α-cells, the activation of these events require higher glucose concentrations in β and also in δ-cells. The stimulus-secretion coupling process and intracellular Ca2+ ([Ca2+]i) dynamics that allow β-cells to secrete is well-accepted. Conversely, the mechanisms that regulate α- and δ-cell secretion are still under study. Here, we will consider the glucose-induced signaling mechanisms in each cell type and the mathematical models that explain Ca2+ dynamics.
The regulation of glycemia depends on the individual action of the different populations that form the islet and their functional interaction. Each islet is composed of a heterogeneous group of 1000–3000 cells, where the predominant cell type is the insulin-releasing β-cell, which coexists with the population of α-cells, responsible for glucagon secretion, and δ-cells, responsible for somatostatin release. While β- and α-cell populations represent, respectively, about 70–80% and 20% of the islet in mice, recent studies in human have demonstrated an increased presence of the latter population (~55% of β-cells versus ~40% of α-cells) (Cabrera et al., 2006). In the case of δ-cells, they can represent a 10% of the islet. Other cell types such as pancreatic polypeptide-producing cells constitute less than 1%. As we will discuss later, the interaction among these cell types depends not only on their proportions within the islet but also on their spatial distribution. The control of glycemia as a result of islet function depends mainly on the direct action of α- and β-cell populations. While β-cells release insulin with increased glucose concentrations, glucagon secretion from α-cells takes place in hypoglycemic conditions (Quesada et al., 2008). However, δ-cells exert an indirect action on glycemia by paracrine mechanisms given that the release of somatostatin inhibits α- and β-cell function (Nadal et al., 1999; Hauge-Evans et al., 2009).
β-cells of pancreatic islets are responsible for the biosynthesis, storage, and release of insulin. Insulin is a peptide hormone that controls blood glucose concentration. A decrease in either insulin release or its peripheral effects (insulin resistance) results in diabetes, a devastating condition suffered by more than 150×106 people worldwide. When nutrients (glucose, amino acids) increase in the blood, insulin is released to the portal system then acting first in the liver and subsequently in peripheral tissues (muscle, adipose tissue). The effect of insulin on specific receptors, activate glucose uptake into the cell and promotes energy storage (glycogen, fat). Under physiological conditions, the blood glucose concentration in fasted mammals, including humans, is maintained at around 5 m mol/l. While lower glucose concentrations (hypoglycaemia) may be lethal for certain cell types (nerve cells), higher concentrations (hyperglycaemia) damage cellular structures and cause long-term complications (rethynopathy, nephropathy, neuropathy, etc.). Blood glucose is continuously and precisely controlled by an integrated group of agents. A broad group of agents (glucagon, catecholamine) increases blood glucose but only insulin decreases it. Thus, no physiological substitutes of the hypoglycaemic factor may act when insulin is not present.
A better knowledge of β-cells is instrumental in order to find new treatments for diabetes or in the attempt to generate β-cell surrogates, one of the central aims of regenerative medicine (Soria et al., 2000; León-Quinto et al., 2004; Vaca et al., 2006). Pancreatic islet cells are also a model for calcium-dependent peptide release. As compared with neurotransmitter release, which last for periods measured in milliseconds, hormone peptide release is a slow process lasting for seconds or even minutes. However, both secretory systems share common mechanisms (Martín et al., 1995a, 1996). The aim of this review is to discuss the experimental and mathematical models that explain how this system works under physiological conditions.
Pancreatic β-cells are blood glucose sensors, which adapt release insulin to the amount of glucose detected. Glucose sensor mechanisms include: (i) a high capacity and low-affinity transporter (Glut-2) acting as an open door that equalizes glucose both sides of the membrane, (ii) a low-affinity isoform of hexokinase, hexokinase IV is also called as glucokinase, (iii) glucose metabolism producing intermediates such as adenosine triphosphate (ATP) and diadenosine polyphosphates (DPs), (iv) a potassium channel (KATP) sensitive to the increase in ATP and DPs, and (v) a voltage-dependent Ca2+ channel (CaV) (Fig. (Fig.1).1). In the case of the β-cell, there is a consensus about the stimulus-secretion coupling model. Electrical activity, [Ca2+]i signals, and consequently, insulin release are all stimulated at high-glucose concentrations. The metabolism of glucose and other nutrients (Bolea et al., 1997), especially at the oxidative level in the mitochondria, leads to a rise of the ATP/adenosine diphosphate (ADP) ratio and diadenosine polyphosphates (Martín et al., 1998), inhibiting ATP-dependent K+ channels (KATP) and depolarizing the β-cell plasma membrane. As a consequence, voltage-dependent Ca2+ channels (CaV) activate, driving extracellular Ca2+ entry (Quesada et al., 2000; Sanchez-Andrés et al., 1988; Santos et al., 1991;Valdeolmillos et al., 1992). Although there are evidences of [Ca2+]i nondependent release (Martín et al., 1995c), it is broadly accepted that this increase in cytosolic Ca2+ is the signal that triggers the exocytotic process and insulin release.
Glucose-induced electrical activity consists in a depolarization followed by repetitive action potentials (calcium spikes) organized in bursts. The duration of these bursts membrane depends on glucose concentration. In order to keep blood glucose at 5 m mol/l, EC50 of the dose-secretion curve has to be close to this figure. In-vitro recorded glucose-induced electrical activity and insulin release have a threshold at about 7 m mol/l and an EC50 value of around 12 m mol/l glucose, far outside the physiological range (5 m mol/l). Only when other nutrients (amino acids) are present the electrical response to glucose is close to the normal physiological level of glucose (Bolea et al., 1997). Previous studies from our group showed that intracellular calcium (in the whole islet) and cell bursting electrical activity (in one single β-cell) are synchronized (Santos et al., 1991). This observation suggest that each calcium wave is due to Ca2+ entering the cells during a depolarized phase of electrical activity and that calcium oscillations occur synchronously across the whole islet tissue. Simultaneous measurements of [Ca2+]i and insulin release resolved pulsatile insulin secretion that paralleled slow [Ca2+]i oscillations (Martín et al., 1995b).
In contrast with neurotransmitter release, in endocrine cells secretion is relatively slow and continues for tens of millisecond after Ca2+ entry through the voltage-dependent Ca2+ channels has ceased, there is a latency between [Ca2+]i and exocytosis and since calcium affinity of the sensor mechanism is lower, a single action potential may not be sufficient to trigger significant secretion and trains of action potentials may be more effective. The properties of the calcium-sensor for glucose-induced insulin secretion may be explored using cell-permeant [Ca2+]i buffers with different kinetics and affinities (Pertusa et al., 1999). Slow Ca2+ buffers [ethylene glycol tetraacetic acid (EGTA) and calcium orange-SN] did not affect glucose-induced insulin release while fast Ca2+ buffers (BAPTA and calcium green-5N) caused a 50% inhibition of the early phase and completely blocked late phase of insulin release. These data are consistent with: (i) the existence of a calcium-sensor with higher affinity than that of neurons and (ii) the existence of two pools of granules: “primed” vesicles colocalized with Ca2+-channels responsible for the first phase of insulin release and “reserve pool” vesicles not colocalized and responsible for the second phase.
Glucose exerts three basic actions on the intracellular calcium activity [Ca2+]i: an increase in extracellular glucose induces an increase beneath the membrane, a decrease in the rest of the cytosol, and an increase in the intranuclear space (Fig. (Fig.2).2). Glucose-induced [Ca2+]i changes not only trigger exocytosis when increases in a restricted space beneath the membrane (Quesada et al., 2000) but also controls gene expression at the nuclear level (Quesada et al., 2002). Furthermore, since high [Ca2+]i levels are toxic for any cell, both mitochondria (than in addition provides energy) and endoplasmic reticulum are responsible for the nutrient-induced Ca2+ sequestration (Valdeolmillos et al., 1992). Mathematical models explaining Ca2+ dynamics should take in account the existence of these pools.
[Ca2+]i changes beneath the cell membrane are those directly responsible for exocytosis. To monitor those changes we developed a confocal spot microscope able to follow an optical signal in a region of 0.6 μm of diameter (Quesada et al., 2000). Glucose metabolism generates [Ca2+]i microgradients that reach the values of 8–10 μmol/l beneath the membrane that rapidly decayed to 0.27 μmol/l at a distance of 2 μm (Fig. (Fig.2).2). Mitochondrial [Ca2+]i uptake and endoplasmic stores are involved in its regulation. The computer simulation of the effect of Ca2+ chelators on [Ca2+]i gradients resulted in similar figures (Pertusa et al., 1999). Conversely, this increase in [Ca2+]i beneath the membrane nutrients decreases it in the cytosol. This second effect is usually masked by the increase when the system has not spatial resolution. Before using spot confocal microscopy we visualized this effect by adding diazoxide (see Fig. 6 of Valdeolmillos et al., 1992). Diazoxide opens KATP channels and keeps the membrane potential clamped at the potassium equilibrium potential. Under these circumstances, addition of glucose decreases [Ca2+]i. Both mitochondria and endoplasmic reticulum participate in the sequestration of cytosolic calcium. The third effect of glucose on [Ca2+]i is an increase in the nucleoplasmic space (Quesada et al., 2002). Patch-clamp recording of the nuclear envelope allowed us to identify an ATP-sensitive K-channel (KATP) with similar properties to that found in the plasma membrane. The blockade of KATP channel with tolbutamide or diadenosine polyphosphates triggers nuclear [Ca2+] transients and induce phosphorilation of the transcription factor CREB (camp response element). Then KATP channels in the nuclear envelope link glucose metabolism, nuclear [Ca2+] signals and nuclear function.
Ca2+ triggered secretion in pancreatic β-cells is known to be a relatively slow process with longer latencies when compared with neurotransmitter release. A plausible explanation for such slow mechanism is the existence of cytosolic barriers (endogenous buffers) that delay the response by stepping down the free calcium transient. The effect of exogenous chelators (Pertusa et al., 1999) supports this possibility. Furthermore, not only nutrients control these cells but blood glucose regulation also relies on multiple levels of hormonal and neuronal control (Quesada et al., 2008). Both inhibitory sympathetic and stimulatory parasympathetic inputs control endocrine pancreas. Cholinergic effects on membrane potential and cytosolic calcium acts as potentiators of glucose (initiator of insulin release). We have previously shown that cholinergic potentiation of insulin release is initiated by an early calcium entry (Sanchez-Andrés et al., 1988). Biphasic responses of β-cells to changes in glucose concentration and the depolarizing effects of the sarcoendoplasmic reticulum calcium ATP-ase pump poison thapsigargin are accounted in a model, which incorporates ICRAC in β-cells (Bertram et al., 1995).
To explain cell behavior physiologists use structures (cell membranes, channels, mitochondrias, etc.) and concentrations of free particles. The structures also have concentrations and densities (n channels per μm2 or 1,000 mitochondria/cell). Since physiological functions need to be quantified we use chemical parameters (affinity, concentrations, etc.) to explore the interactions between the free particles or between the free particles and structures. Mathematical models describing such processes could be classified in continuous models and stochastic models. Stochastic models are, in general, suitable for discrete and/or fast processes or for systems presenting elements with irregular geometry such as, for instance, the distribution of channels and vesicles on the membrane surface of neuroendocrine cells (Gil et al. 2000; Segura et al. 2000). On the other hand, continuous models, based on the solution of systems of differential equations, are more appropriated for slow and/or regular systems. This is the case of the secretory response in pancreatic cells.
A significant number of papers covering different aspects of modeling of pancreatic β-cells can be found in scientific literature: Diederichs (2006, 2008), Fridlyand et al. (2003, 2005), Meyer-Hermann (2007), Zhan et al. (2007), Boutayeb and Chetouani (2006), and Bertram et al. (2007, 2009) were just some of the examples of recent papers on modeling electrical activity, glucose metabolism, and insulin secretion in β-cells. Specifically, Diederichs (2006) focused on the study of the glucose metabolism of the pancreatic β-cell, proposing a mathematical model that includes membrane fluxes and currents, as well as metabolic processes of the beta-cell during rest and activation, including oscillations. Fridlyand et al. (2003) presented a model for β-cells that include an extended variety of ionic channels and pumps as well as other calcium sequestration mechanisms in order to investigate the possible role of K+ channels, Na+, and IP3 in regulating beta-cell Ca2+ oscillations. In Fridlyand et al. (2005), the authors used the model presented in Fridlyand et al. (2003) to relate glucose consumption, nucleotide pool concentration, respiration, Ca2+ fluxes, and KATP channel activity to study high-glucose-induced [Ca2+]i oscillations. In Meyer-Hermann (2007), a model based on a bottom-up approach, which includes modeling mechanisms using the data from single protein level or conductance measurements, is presented. Zhan et al. (2007) considered a mathematical model for beta-cells describing the effects of both glucose and IP3 concentrations on membrane potentials. Boutayeb and Chetouani (2006) presented a global overview of mathematical models dealing with many aspects of diabetes and using various tools, including glucose and insulin dynamics, management and complications prevention, cost and cost effectiveness of strategies, and epidemiology of diabetes in general. Finally, Bertram et al. (2007) and Bertram et al. (2009) were just a two of the recent examples of many publications about this theoretical group on beta-cell modeling and application to different aspects of beta-cell regulation. The group also has an open-source web page for the software: www.math.fsu.edu/~bertram/software/islet. In contrast with the large number of papers on the modeling of beta-cells, there are very few theoretical studies on alpha cells, probably due to the limited number of experimental data available (Diderichsen and Göpel 2006; González Velez et al. 2009). The scarcity of models is even more noticeable in the case of pancreatic delta-cells for which theoretical models are not available at the level of single cell but limited to pancreatic islet studies (Jo et al. 2009).
Changes in the intracellular Ca2+ concentration are a key element in the signaling pathway that results in hormone release not only in β-cells but also in α- and δ-cells. These Ca2+ signals are a consequence of the characteristic set of channels and electrical activity of each cell type (Kanno et al., 2002). Although there is not an established model for α- and δ-cell function, it is accepted in both cases that glucose concentrations modulate their electrical activity, Ca2+ signals, and hormone secretion (Gromada et al., 2004). In the light of evidences accumulated in the last years, it is assumed that δ-cells operate in a very similar scheme as β-cells (Kanno et al., 2002) increases in extracellular glucose levels enhance electrical and Ca2+ signaling activity, and thus, somatostatin release. Since these cells possess KATP channels as well, it has been proposed that they link metabolic changes to an electrical and Ca2+ response by producing membrane depolarization, which leads to the generation of action potentials mediated by Ca2+ and Na+ channels (Gopel et al., 2000a; Kanno et al., 2002). This activation promotes Ca2+ entry and granule exocytosis. These cells are activated at glucose concentrations as low as 3 m mol/l (Nadal et al., 1999). This increased response to glucose compared to β-cells has been explained by the smaller density of KATP channels in the δ-cells since in this case, the membrane depolarization required to reach the threshold to generate action potentials would take place before a complete inhibition of KATP channels occurs (Kanno et al., 2002). Thus, glucose seems to exert a direct effect on the regulation of somatostatin secretion. Actually, glucose modulates Ca2+ signals in isolated δ-cells (Berts et al., 1996). In these conditions, they are not exposed to the influence of paracrine factors. In addition, recent studies using redox confocal microscopy in intact islets have shown that glucose enhances the mitochondrial metabolism in δ-cells (Quesada et al., 2006b), further supporting the link between glucose metabolism and somatostatin release by means of KATP channels. Thus, the stimulus-secretion coupling mechanisms in response to glucose in δ-cells should be very similar as those working in β-cells. Recently, it has been reported that a KATP channel independent pathway is also operating in δ-cells for glucose concentrations higher that 10 m mol/l (Zhang et al., 2007). This pathway involves a process of Ca2+-induced Ca2+-release. Additionally, the control of δ-cell secretion is further subjected to paracrine and neural communication. However, little is known about this kind of regulation at the cellular level.
In contrast to the case of β- and δ-cells, electrical activity, Ca2+ signals, and hormone release are stimulated at low glucose levels in the α-cell population while all these events are inhibited when glucose concentrations are elevated (Gromada et al., 2004; Quesada et al., 2008). Thus, they have an opposite physiological response to that of β- and δ-cells. So far, there is not a consensus model to explain glucagon secretion in response to glucose changes. Moreover, α-cells are notably regulated by other key levels of control, particularly paracrine mechanisms (Ishihara et al., 2003), which complicates the discrimination of a direct action of glucose in the inhibition of glucagon release at high-glucose concentrations. In any case, a model describing the stimulus-secretion coupling for α-cells has been proposed (Ashcroft, 2000; Gopel et al., 2000b; Gromada et al., 2004). This model reconciles recent findings in intact mouse islets and isolated cells but leaves unanswered some other aspects. According to this model, KATP channels, which are present in α-cells as well, are also responsible for changes in the membrane potential. At low glucose concentrations, the negative membrane potential in these cells is situated within a range that allows an electrical activity based on action potentials of T-type voltage-dependent Ca2+ channels. These channels drive the membrane potential to more positive values, allowing the activation of Na+ and N-type voltage-dependent Ca2+ channels. Repolarization would take place because of A-type K+ channels activation. Ca2+ entering through Ca2+ channels would be responsible for triggering granule exocytosis. An increase in the extracellular glucose levels would produce a rise in the ATP/ADP ratio as a consequence of glucose intracellular metabolism, blocking KATP channels. This inhibition would depolarize the plasma membrane to potential values that inactivate all the abovementioned currents and channels that sustain action potentials, consequently decreasing electrical activity, Ca2+ signaling, and glucagon release (Gromada et al., 2004). Later studies have demonstrated that glucagon secretion is stimulated within a window of intermediate KATP channel activity, which leads to an appropriate range of membrane potential that drives a sustained action potential firing (MacDonald et al., 2007). Thus, this model propose a direct action of glucose and a key role of KATP channels to couple metabolic changes to ionic currents and Ca2+ signals that control secretion.
However, several studies have reported important differences between α- and β-cells that do not fit in the abovementioned model. Although both cell types possess different glucose transporters, it has been demonstrated that glucose transport is not a limiting factor in the metabolism of this sugar (Gorus et al., 1984; Heimberg et al., 1995, 1996). It seems that the main differences take place at the mitochondrial level. Using redox confocal microscopy to monitor metabolic changes and immunochemistry to identify the different islet cell types, it has been reported that α-cells exhibit a much lower activation of the mitochondrial aerobic metabolism in response to glucose (Fig. (Fig.3)3) (Quesada et al., 2006b). Actually, several biochemical studies in islet cells separated by cell sorting indicate a higher level of anaerobic metabolism in α-cells in contrast to the elevated mitochondrial metabolic activity of β-cells (Schuit et al., 1997). This idea is also supported by investigations that indicate that the ratio of lactate dehydrogenase/mitochondrial glycerol phosphate dehydrogenase is low in the β-cell, favoring the mitochondrial oxidation of glucose while this ratio is higher in non-β-cells (Sekine et al., 1994). Thus, it seems that α-cells exhibit a very low coupling between the glycolytic events in the cytosol and ATP synthesis in the mitochondrial respiration. All these information about the α-cell glucose metabolism agrees with the fact that ATP/ADP changes in response to glucose are almost invariable in α-cells compared to the situation of β-cells (Detimary et al., 1998). Therefore, there are some questionable aspects in the mentioned model for the α-cell, especially those concerning the regulation of the KATP channel activity by an increase in the ATP/ADP ratio. There are probably some other additional mechanisms regulating KATP channels, and thus, this model will probably require the incorporation of new elements. In this regard, in addition to a direct effect of glucose, several paracrine mechanisms have an important effect in the inhibition of glucagon release such as insulin, zinc, somatostatin, and γ-aminobutyric acid, mainly by interaction with electrical activity (Ishihara et al., 2003; Ravier and Rutter, 2005; Wendt et al., 2004; Quoix et al., 2009). Moreover, it has been proposed that a store-operated membrane conductance regulates α-cell electrical activity in response to glucose independently of KATP channels (Liu et al., 2004; Vieira et al., 2007). Recent studies also demonstrate that the hyperpolarization-activated cyclic nucleotide-gated channels are present in α-cells and play a role in the regulation of glucagon secretion by modulating the activity of Na+ and Ca2+ channels (Zhang et al., 2008). All these different mechanisms are most likely operating in the regulation of glucagon release. In the light of new evidences, more studies will be required to establish a model for the stimulus-secretion coupling in the α-cell.
The majority of studies about islet function have been focused on β-cell and insulin release. However, little is known about the physiology of other islet cell populations despite their relevance in glucose homeostasis. This lack of information is the result of several factors (Quesada et al., 2008). One of them is the scarcity of the different cell types versus the β-cell predominance in the islet. Additionally, because of a limited characterization of these non-β populations, individual and specific physiological patterns for each cell type were absent. Finally, most of the available approaches to investigate islet physiology presented technical limitations to study individually each cell type. Although the information about non-β-cells from studies with cell lines or isolated cells are relevant, several physiological differences have been found when these cells are within the islet. Actually, several “in vivo” studies have evidenced that the intact islet as a study model is closer to the physiological scenario (Sanchez-Andres et al., 1995). These discrepancies result from the critical importance of the intercellular communication and paracrine regulations within the islet an aspect that should be included in future mathematical models. These reasons have motivated the exploration of the physiological behavior of the different cell types in the intact islet. In the last years, several groups have characterized the electrophysiological properties and Ca2+ signals in response to glucose of the main cell types in the intact islet (Gopel et al., 2000a; Nadal et al., 1999; Quesada et al., 2006a).
The study of individual Ca2+ responses with conventional microscopy is impractical as a result of the out-of-focus information. With this technique, the fluorescence emitted from Ca2+-sensitive probes arrive to the detector from the focal plane but also from adjacent planes. Since a large fraction of the islet is composed by the β-cell population, the fluorescence from multiple planes will be dominated by the β-cell signal, and thus, the response of α- and δ-cells, which are less abundant, will not be discriminated. This limitation can be overcome using confocal microscopy (Fig. (Fig.4)4) (Nadal et al., 1999). With this technique, fluorescence is only monitored from a thin optical section and thus, light mainly comes from the focal plane without the interference of information from out-of-focus planes (Quesada et al., 2000). In the three cell types α, β, and δ, intracellular Ca2+ increases are the signal that triggers exocytosis and hormone secretion. For a long time, the detection of the characteristic Ca2+ response of each cell type within the islet was not feasible. Confocal microscopy allowed the detection of different cell-specific Ca2+ signals in response to glucose (Fig. (Fig.2),2), which were correlated with cell types by immunochemistry (Nadal et al., 1999; Quesada et al., 2006a). Using these methods, it was reported that there is a cell population that is silent at 3 mM glucose and exhibits an intracellular Ca2+ increase followed by a train of oscillations when glucose is elevated to 11 m mol/l. This is the typical response of the β-cell. However, two groups of cells have an oscillatory Ca2+ signal in 3 m mol/l. When glucose is elevated to 11 m mol/l, a group corresponding to the α-cell population becomes silent while another group identified as δ-cells remains without significant changes in frequency (Fig. (Fig.4).4). At 0.5 m mol/l glucose, only α-cells display Ca2+ signaling. The behavior of all these Ca2+ signals correlates with the patterns of electrical and secretory activity in response to glucose for each cell type (Kanno et al., 2002). Those signaling patterns were demonstrated in mice and, more recently, in humans (Nadal et al., 1999; Quesada et al., 2006a).
In mice, all the oscillatory Ca2+ signals from individual β-cells are synchronized throughout the islet (Nadal et al., 1999; Santos et al., 1991). This communication also occurs at the electrical level and is a consequence of the high degree of coupling among β-cells, which is mainly mediated by gap-junctions of connexin 36 (Quesada et al., 2003; Serre-Beinier et al., 2000). This synchronization of individual Ca2+ signals would explain that Ca2+ oscillations can be recorded from the islet with conventional fluorescence microscopy. With nonsynchronic individual responses, the oscillations would be cancelled out. It has been indicated that this signal synchronization would lead to a more vigorous insulin secretion (Vozzi et al., 1995). In humans, however, coupling among individual Ca2+ signals can only be found in small clusters of cells but not in the whole islet (Quesada et al., 2006a). The significance of this is still unknown. In any case, this functional organization agrees with the spatial cell organization in the islet. While in mice the majority of β-cells are close together in the islet center and are surrounded by a peripheral layer of non-β-cells, in humans, β and non-β-cells are distributed individually or in small clusters (Cabrera et al., 2006). In the case of α- and δ-cells, they work as individual units in terms of signaling in both mice and humans. Electrophysiological techniques as well as studies of their Ca2+ signals demonstrated no functional coupling in these cells. So far, mathematical models provided plausible explanations for glucose-induced Ca2+ signaling and insulin release. However, pancreatic β-cells are integrated in islets interacting with other cell types. In addition, both in rodents and in humans β-cells are connected through gap-junctions, which allow fluxes of Ca2+ between cells, as demonstrated in the recovery after bleaching experiments (Quesada et al., 2003). More than 20 years ago Sherman et al. (1988) suggested an attractive explanation by which “channel sharing” in clusters of cells tightly coupled by gap junctions could convert chaotic behavior of single cells into regular bursting in islets. Then, high conductance gap junction coupling, which converts the cluster into a single “supercell” determines the minimal size for the cluster in 10–50 cells (Sherman and Rinzel, 1991). Since depending of their size or species involved islets may contain more than one synchronized cluster, a more recent model (De Vries et al., 1998) analyzed this behavior. Gap junctions not only modulate electrical coupling but also functional fluxes of calcium diffusion concluding that is not necessary for islet synchronization but not excluding other physiological roles, e.g., glycolytic oscillations. More recent mathematical models have analyzed the beneficial effects of intercellular interactions between pancreatic islet cells in blood glucose regulation (Jo et al., 2009) or the role of intrapancreatic ganglia islet synchronization mechanisms (Fendler et al., 2009). Kang et al. (2008) presents a simple islet model consisting of alpha-, beta-, and delta-cells, where clusters of beta-cells dominate glucose regulation. They connect the microscopic bursting mechanism and the macroscopic blood glucose regulation of the body. Finally, in Jo et al. (2009) a mathematical model for an islet consisting of two-state alpha-, beta-, and delta-cells is presented in order to analyze cell-to-cell variations in response to external glucose concentrations as well as glucose dynamics, which depends on insulin and glucagon. The combination of transgenic technology (connexin 36 knock-out mice) with high-speed imaging and theoretical modeling (Benninger et al., 2008) uncovered the mechanisms underlying emergent properties of islets such as electrical bursting. Future mathematical models should incorporate all these observations then giving a more appropriate description of the behavior of cells at the single cell level and at the islet (micro organ) level.
The authors are supported by the Fundación Progreso y Salud, Consejería de Salud, Junta de Andalucía (Grant No. PI-0022/2008), the Spanish Ministry of Science and Innovation (Grant Nos. BFU2007-67607 and BFU2008-01492), and the Red TERCEL (Grant No. RD06/0010/0025). CIBERDEM is an initiative of Instituto de Salud Carlos III.