Bursting and calcium oscillations in pancreatic β-cells: specific pacemakers for specific mechanisms

Abstract

Oscillatory phenomenon in electrical activity and cytoplasmic calcium concentration in response to glucose are intimately connected to multiple key aspects of pancreatic β-cell physiology. However, there is no single model for oscillatory mechanisms in these cells. We set out to identify possible pacemaker candidates for burst activity and cytoplasmic Ca²⁺ oscillations in these cells by analyzing published hypotheses, their corresponding mathematical models, and relevant experimental data. We found that although no single pacemaker can account for the variety of oscillatory phenomena in β-cells, at least several separate mechanisms can underlie specific kinds of oscillations. According to our analysis, slowly activating Ca²⁺-sensitive K⁺ channels can be responsible for very fast Ca²⁺ oscillations; changes in the ATP/ADP ratio and in the endoplasmic reticulum calcium concentration can be pacemakers for both fast bursts and cytoplasmic calcium oscillations, and cyclical cytoplasmic Na⁺ changes may underlie patterning of slow calcium oscillations. However, these mechanisms still lack direct confirmation, and their potential interactions raises new issues. Further studies supported by improved mathematical models are necessary to understand oscillatory phenomena in β-cell physiology.

in pancreatic β-cells, glucose-stimulated insulin secretion is mediated by an elevated cytosolic free calcium concentration ([Ca²⁺]_c). This results from calcium influx through voltage-dependent Ca²⁺ channels (VDCCs) located in the plasma membrane (PM), which open in response to secretagogues, primarily glucose. Stimulation-secretion coupling in β-cells is different from most other cell types, because instead of being mediated by receptor binding, glucose must be transported into the cytoplasm and metabolized.

Glucose initiates changes in the PM potential via an increase in the cytoplasmic [ATP]/[ADP] ratio derived from glycolysis and oxidative phosphorylation. This results in closure of ATP-sensitive K⁺ (K_ATP) channels. Closure of these channels leads to PM depolarization up to a threshold potential, causing the cell to move from quiescence to initiate electrical activity and VDCC opening. Ca²⁺ influx through VDCCs leads to increased [Ca²⁺]_c, which is a key signal in the initiation of insulin secretion from the pancreatic β-cells. Consensus mechanisms of glucose-induced PM, cytosolic, and mitochondrial processes are summarized in Fig. 1 (for recent review, see Refs. 71, 74, 76, and 140).

Fig. 1.

General scheme of the main processes involved in bursting and intracellular Ca²⁺ oscillations in pancreatic β-cells. Plasma membrane currents: voltage-dependent Ca²⁺ current (I_VCa), calcium pump current (I_Cap), Na⁺/Ca²⁺ exchange current (I_NaCa), Ca²⁺ release-activated nonselective cation current (I_SOC), inward Na⁺ currents (I_Na), sodium-potassium pump current (I_NaK), slow Ca²⁺-sensitive K⁺ current (I_KCas), delayed rectifying K⁺ current (I_KDr), small conductance Ca²⁺-activated K⁺ current (I_KCa), ATP-sensitive K⁺ current (I_KATp). k_sg is a coefficient of the sequestration rate of Ca²⁺ by the secretory granules. SERCA is a calcium pump in endoplasmic reticulum (ER), and Ca²⁺ leaks from ER throw into inositol 1,4,5-triphosphate (IP₃) receptor (IP₃R) and ryanodine receptor (RyaR). ATP is the free cytosolic form of ATP, and ADP_f is free cytosolic ADP. Signals originating from fuel metabolism increase cytosolic calcium. F6P, fructose 6-phosphate. Synthesis and degradation of cAMP are catalyzed by adenylyl cyclase (AC) and phosphodiesterase (PDE), respectively. AC was activated by stimulatory G protein and Ca²⁺ and deactivated by inhibited G protein, whereas PDE activity was enhanced by Ca²⁺. Synthesis of IP₃ is catalyzed by phospholipase C (PLC). PLC was activated by stimulatory G protein. Solid lines indicate fluxes, and dashed lines indicate inhibitory or stimulatory influences on currents or fluxes.

The β-cell membrane is hyperpolarized to a resting potential of about −60 mV at low glucose levels (∼3–6 mM). Dean and Matthews (32, 33) showed that when glucose rises the PM depolarizes and then generates an electrical activity organized into slow depolarizing waves, called bursting, with a plateau from which action potentials (APs) rapidly fire. Bursts (or active phases) are separated by quiescent (resting) periods at potentials below the AP threshold (Fig. 2). This bursting process results from the metabolic processes and electrical activity of ion channels localized on the β-cell PM and regenerates as long as the glucose concentration is elevated. Rapid depolarization at the beginning of a burst activates VDCCs, whereas the fast repolarization at the end of a burst results in inactivation of VDCCs. This leads to synchronized cyclical spike-burst activity and corresponding cytoplasmic Ca²⁺ oscillations in response to a rise in extracellular glucose that ultimately determine the oscillatory nature of insulin secretion (71, 76, 140).

Fig. 2.

Temporal correlation between membrane potential (MP) and cytosolic free calcium concentration ([Ca²⁺]_c) oscillations (adapted from Ref. 11). Islets from wild mice were perifused with a solution containing 8 (C and D) or 10 mM glucose(A and B) as indicated. The MP was measured in a β-cell within an islet, and [Ca²⁺]_c was measured simultaneously in the same islet.

Increased glucose concentration induces several types of cyclical spike-burst activity and [Ca²⁺]_c oscillations in β-cells with period ranges from several seconds to several minutes (11, 19, 71). We have divided these oscillations into several classes: 1) bursts and [Ca²⁺]_c oscillations with a period of several seconds we term “ultrafast”; 2) oscillations with a period from 10 to 60 s, denoted as fast (Fig. 2A); 3) Ca²⁺ oscillations with a period ranging from 1 to several minutes and long bursts (Fig. 2B), designated as slow; and 4) mixed (or compound) [Ca²⁺]_c oscillations, characterized by ultrafast or fast [Ca²⁺]_c oscillations superposed on slow oscillations; this is due to periodic bursts, called compound bursting (Fig. 2, C and D) (17).

Ultrafast oscillations can be observed in isolated single β-cells (91, 150). Fast electrical bursting and cytoplasmic Ca²⁺ oscillations are usually observed in isolated mouse islets (11, 57). Slow and mixed cytoplasmic Ca²⁺ oscillations were found in single cells, β-cells clusters, and isolated islets at stimulating glucose concentrations (10, 11, 57, 70, 99). Slow and mixed oscillations most likely constitute a physiological oscillatory pattern in β-cells (57, 140).

Gap junctions are responsible for tight electrical coupling at least in mouse islets. This leads to the synchronization of bursting and calcium oscillations in an islet (12, 94, 144). For this reason, as a first approximation we can consider the islet as a community of coupled β-cells with similar electrical behavior.

Experimental data and theoretical concepts support the idea that the AP bursts and the corresponding [Ca²⁺]_c oscillations reflect a periodic depolarization of the PM (7, 11, 37, 48). Whereas a depolarizing component predominates at the beginning of a burst, a progressive increase in outward and/or decrease in inward cation currents take place during the burst, leading to a repolarization and burst termination. Slow depolarization in a resting (silent) phase leads again to a burst initiation (7, 48). Experimental data and theoretical analysis agree that, after the large depolarizing effect following K_ATP channel closure, the subsequent generation and termination of the bursts can be maintained by small cyclical changes of any PM current (9, 48, 132, 133).

Consequently, initiation and termination of bursts can be connected directly with activation and deactivation of different electrogenic channels, pumps, or exchangers, which can be modulated in turn by numerous intracellular components. Therefore, it is not surprising that a variety of different proposals and corresponding mathematical models have been advanced, underlining the nature of these components in the β-cells. A list of the compounds to be considered here is shown in Table 1. Several of these modulators can act simultaneously, generating AP bursts and corresponding [Ca²⁺]_c oscillations. However, the exact physiological variables and their underlying mechanisms that drive bursting or Ca²⁺ oscillations in β-cells remain uncertain. Here, we attempt to identify the possible pacemaker candidates for burst activity and cytoplasmic Ca²⁺ oscillations in the pancreatic β-cells.

Table 1.

Candidates for pacemakers regulating β-cell bursting and [Ca²⁺]_c oscillations

Compound or Process	Reference No(s).
Ca2+-dependent K+ current	8, 26
Slowly activated Ca2+-sensitive K+ current	58, 59
Voltage-gated Ca2+ current	28, 88
[ATP]/[ADP] ratio	3, 126, 133
Glycolytic oscillations	17, 19, 142, 145
[Ca2+] in endoplasmic reticulum	20, 25, 51, 125, 148
Intracellular Na+	51, 93
Inositol (1,4,5)-trisphosphate	This article
cAMP	47
Mitochondrial metabolic oscillations	14, 101

[Ca²⁺]_c, cytosolic free calcium concentration.

We define a pacemaker as a component of some pacemaker mechanism that serves to establish a rhythmic physiological activity inside a cell. To distinguish likely pacemaker proposals from less likely hypotheses, we will reject a hypothesis that some component is a part of a pacemaker mechanism if Ca²⁺ oscillations (or the AP bursting pattern) persist even though the proposed pacemaker mechanism was blocked in an experiment. We will term this condition the main criterion for the rejection of a proposed pacemaker mechanism. Disappearance or a change of bursting or Ca²⁺ oscillation behavior under the influence of an agent or process we will term a “modulation,” and an agent that creates it we will term a “modulator.”

A pacemaker component can also act as a modulator with a certain concentration dependence, depending on its activity or experimental conditions. However, a modulator in this article is a component that is not a part of a proposed pacemaker mechanism. Several examples of such pacemakers and modulators will be considered.

We performed an analysis of the existing hypothesis, experimental data, and the corresponding mathematical models and compared the simulations with the experimental results. In this way we tested both the validity of the initial hypotheses and the models. Furthermore, predictions for unmeasured quantities could be made, and the model could then be tested. However, many of the mechanisms that could control electrical activity and Ca²⁺ handling have not been adequately experimentally characterized in pancreatic β-cells, so we can consider them only briefly. Although most of the analyzed experimental work employed mouse islets with some studies in other species, it is difficult to generalize given the different existing animal and insulinoma models of pancreatic β-cells. We have emphasized primary tissues where possible.

There are several reasons why a clear understanding of internal oscillatory mechanisms in β-cells is worth pursuing. First, bursting and [Ca²⁺]_c oscillations are readily observable phenomena in cell culture and in isolated mouse islets. Numerous hormones and small molecule agonists as well as toxins can modulate oscillatory behavior. The study of the changes in bursting or [Ca²⁺]_c oscillations can provide a flexible approach to understand the action of specific modulators on the pancreatic β-cell. However, this approach requires a comprehensive knowledge of the pacemaker mechanisms of these oscillations.

We consider here primarily the internal pacemaker mechanisms for individual β-cells that represent behavior of isolated cells or the electrically synchronized islet. A further consideration of the oscillations in insulin secretion is beyond the scope of the present article. However, islet Ca²⁺ oscillations appear to be the framework behind the phenomenon of pulsatile insulin release observed in vivo (71, 112, 117, 140). Interestingly, disruptions in rhythmic function may be an important early biomarker of islet dysfunction leading to diabetes in animal models (72, 79, 140) and man (116). Although pulsatility appears to be a natural function of islets both in vivo and in vitro, this emphasizes the need for careful study of the oscillatory phenomena on the cellular level.

EXPERIMENTAL PROCEDURES AND MODELING

Ca²⁺ imaging measurements of relative cytoplasmic Ca²⁺ concentrations are presented as the 340/380 nm fluorescence excitation ratio obtained with the indicator fura-2, as described previously (137). Fura-2 was from Invitrogen (Carlsbad, CA). All other chemicals were from Sigma (St. Louis, MO).

The “complex model” of processes (51) was used for the following simulations, excluding the model for slowly activated Ca²⁺-sensitive K⁺ channels, that are described in the appendix. This model is available for direct simulation on the web site “Virtual Cell” (www.nrcam.uchc.edu) in the “MathModel Database” on the “math workspace” in the library “Fridlyand” with name “Chicago 1.”

RESULTS AND DISCUSSION

We can distinguish two main types of mechanisms underlying bursting and [Ca²⁺]_c oscillations.

Mechanisms With Ca²⁺ Feedback Effect

In this case, a fast increase of [Ca²⁺]_c during an active phase should activate (or deactivate) some processes, leading to slow changes in the intermediate pacemaker components. Slow changes of these components should gradually lead to a membrane repolarization down to the threshold potential for spike activity, shutting down the spikes and inactivating the VDCCs. Then Ca²⁺ influx decreases sharply, leading to decreased PM potential and a resting phase. Processes occur in opposite directions during a resting (repolarized) phase. The [Ca²⁺]_c decreases because cytoplasmic Ca²⁺ is pumped into the extracellular space and into the endoplasmic reticulum (ER) by specific ATP-consuming Ca²⁺ pumps. Decreases in [Ca²⁺]_c deactivate (or activate) specific processes of the slow changes in the intermediate pacemaker components, leading to PM depolarization up to the threshold potential, reentry of spike activity, Ca²⁺ influx, and an active phase.

Ca²⁺-Independent Mechanisms

This type of mechanism was suggested for oscillations such as glycolytic oscillations that can lead to oscillations in the [ATP]/[ADP] ratio changing the conductance of K_ATP channels (19, 142). In this case, Ca²⁺-independent mechanisms create the oscillations of compounds that can lead to cyclic changes in PM potential, generating the bursts and corresponding [Ca²⁺]_c oscillations (101, 142).

[Ca²⁺]_c as the Pacemaker Component and Ca²⁺-Dependent K⁺ Channels

Fast-activated Ca²⁺-dependent K⁺ (SK1–3 and BK) channels.

Initially, a mechanism with [Ca²⁺]_c as the pacemaker component and a Ca²⁺ feedback effect was proposed for burst generation and [Ca²⁺]_c oscillations via cyclic activation of a Ca²⁺-dependent K⁺ current. According to this hypothesis, Ca²⁺ influx during the active phase would cause a slow rise in [Ca²⁺]_c, which activates fast Ca²⁺-dependent K⁺ (K_Ca) channels repolarizing the PM potential down to a threshold potential. This would shut down AP spiking and inactivate VDCCs, leading to a resting phase. A decrease in the [Ca²⁺]_c level during the resting period would lead to decreased current through K_Ca channels, PM depolarization, and burst initiation. According to this proposal, a cyclical change in [Ca²⁺]_c is a candidate islet pacemaker for burst behavior and [Ca²⁺]_c oscillations per se (8). Corresponding mathematical models have incorporated the cyclical activation and deactivation of K_Ca channels by bursting-induced changes in [Ca²⁺]_c (26).

The principle difficulty with this hypothesis is that [Ca²⁺]_c should slowly increase during a period of depolarization, leading to a slowly increasing current through K_Ca channels and then to a repolarization only at the end of a burst period. However, subsequent Ca²⁺ imaging data indicate that the time for the [Ca²⁺]_c increase is short relative to the oscillation period (129). Another problem with this hypothesis was that apamin, a K_Ca (SK1–3) channel blocker, does not change mouse islet bursting or alter β-cell K⁺ currents in normal mouse islets (4, 58, 59, 96). Slow [Ca²⁺]_c oscillations were also observed in the presence of 10–20 mM tetraethylammonium (TEA), i.e., in conditions when TEA-sensitive K_Ca channels should be nearly completely blocked (124, 125). For these reasons, this mechanism for generation of [Ca²⁺]_c oscillation was ruled out (see Refs. 125 and 132). Despite these issues, mathematical models for β-cells continue to appear with slow [Ca²⁺]_c oscillations on the basis of this hypothesis (36, 108).

However, although here we focus on pacemaker mechanisms, some changes in channel conductance may change PM polarization such that oscillation behavior can be modified. For this reason, although K_Ca (SK1–3 and BK) channels do not directly take part in the creation of the pacemaker mechanism, their inhibition can contribute to PM depolarization and lead to changes in the oscillatory behavior (139).

Slow-activated Ca²⁺-sensitive K⁺ channels.

A specific, slowly activated TEA-insensitive Ca²⁺-sensitive K⁺ current (SK-like current) may underlie the mechanisms of oscillations (58, 59). This current activates gradually with a time constant of 2.3 s and inactivates with a time constant of 6.5 s (59). It was assumed that this current regulates the duration of intervals between successive AP bursts observed when β-cells are exposed to moderately elevated glucose (59, 151).

We developed a special mathematical model to simulate this proposal (see appendix). Here, fast K_Ca channels (SK1–3 and BK; current I_KCa in Fig. 1) as well as slowly activated K_Ca channels (SK-like current; I_KCas in Fig. 1), which according to Düfer et al. (38) can be represented partly by SK4 channels, were modeled, and the time constants for activation gating variables reported by Gopel et al. (59) were used for modeling of I_KCas. The behavior of [Ca²⁺]_c, V_m, and the I_KCas is shown in Fig. 3. The reversible slow cyclical changes in I_KCas were obtained, leading to bursting behavior and [Ca²⁺]_c oscillations. The simulated [Ca²⁺]_c increased rapidly in the beginning of burst period, stabilized by activation of the fast K_Ca (SK1–3) channels and kept at a high level during this period. I_KCas increased during a burst period, leading to repolarization, termination of the spikes, and transition to a silent phase. Inhibition of I_KCas during a resting period led to repolarization and reentry to an active phase.

Fig. 3.

Slow-activated Ca²⁺-sensitive K⁺ channels as pacemaker for ultrafast burst and [Ca²⁺]_c oscillations. Burst behavior of the membrane potential (V_P; A), the oscillation pattern of [Ca²⁺]_c (B), and slow Ca²⁺-sensitive K⁺ current (I_KCas; C). Simulations were made according the model from the appendix.

However, the simulated bursts and [Ca²⁺]_c oscillations had a period of several seconds and corresponded well only to ultrafast bursting and [Ca²⁺]_c oscillations. Therefore, the experimental data on the dynamics of this slow SK-like current and our simulation support the idea that ultrafast bursts and Ca²⁺ oscillations can be driven by an interaction of these slow-activated K_Ca channel parameters with [Ca²⁺]_c changes.

Cyclic Changes in K_ATP Channel Conductance: The Pacemaker Role of [ATP]/[ADP] Ratios

K_ATP channels contribute the predominant β-cell K⁺ conductance in resting low-glucose conditions. Following glucose entry and metabolism, their conductance is greatly diminished. However, the small remaining K_ATP conductance in elevated glucose is still comparable with the conductances of other channels, exchangers, and pumps (133). For this reason, cyclical changes in the K_ATP channel conductance have been proposed as a possible mechanism underlying oscillatory behavior of β-cells (3, 68, 110, 126, 133). In this conceptualization, the [ATP]/[ADP] ratio decreases slowly when [Ca²⁺]_c increases during a burst, leading to a slow opening of K_ATP channels and subsequent PM repolarization as the K⁺ efflux increases. Actually, the opposite should occur since the [ATP]/[ADP] ratio gradually increases during a silent phase in the presence of stimulatory glucose, when [Ca²⁺]_c decreases, leading to the closing of K_ATP channels, PM depolarization, and burst initiation (34, 86, 87, 103–105, 126). According to our classification, this is a mechanism with a Ca²⁺ feedback effect, whereas the [ATP]/[ADP] ratio is a pacemaker “compound.” Several mechanisms and corresponding mathematical models for the [ATP]/[ADP] ratio changes and [Ca²⁺]_c oscillations have been proposed.

An empirical equation for [ATP]/[ADP] ratio changes was introduced in an early mathematical model (133) such that [ATP]/[ADP] ratio slowly decreased as [Ca²⁺]_c increased and increased with decreased [Ca²⁺]_c. Oscillatory changes in the [ATP]/[ADP] ratio, K_ATP channel conductance, bursting, and [Ca²⁺]_c oscillations were simulated using this model (17, 18, 133).

Mechanistic hypotheses underlying the possible influence of [Ca²⁺]_c changes on the [ATP]/[ADP] ratio have also been proposed. Decreased [ATP]/[ADP] ratio during [Ca²⁺]_c rise and an increase of this ratio with [Ca²⁺]_c decline could result from either stimulated ATP hydrolysis or inhibition of ATP production.

[Ca²⁺]_c-activated ATP consumption.

Ca²⁺ activates Ca²⁺ pumps on the PM and in the ER as well and certain other intracellular reactions that use ATP. In this case, increased [Ca²⁺]_c can increase ATP consumption and decrease the [ATP]/[ADP] ratio. An opposite process that increases the [ATP/ADP] ratio due to a decline in ATP consumption can also occur with decreased [Ca²⁺]_c (34, 51).

We illustrated this mechanism using our mathematical model (51) that includes a simulation of ATP consumption due to the work of Ca²⁺ pumps (PM and ER) as well as the Ca²⁺-activated ATP consumption in some cytoplasmic processes. The simulated phase relations for the conditions when the free ADP (and corresponding ATP) concentration in the cytoplasm is the only slow pacemaker parameter in this complex model are shown in Fig. 4. Other slow parameters such as cytoplasmic concentrations of Na⁺, inositol 1,4,5-triphosphate (IP₃), and Ca²⁺ in ER ([Ca²⁺]_ER) were frozen at constant levels, blocking other possible mechanisms underlying Ca²⁺ oscillations in this model. A Ca²⁺ feedback effect model was responsible for bursting and Ca²⁺ oscillations; a rise in [Ca²⁺]_c during the active phase leads to increased Ca²⁺-activated ATP consumption in the cytoplasm and by Ca²⁺ pumps. As this occurs, free [ADP] increases (and the [ATP]/[ADP] ratio decreases), slowly opening K_ATP channels. This in turn leads to a slowly increasing I_KATP and PM repolarization, damping potential spikes and leading to decreased [Ca²⁺]_c. These processes have the opposite direction after decreased [Ca²⁺]_c in the resting period. K_Ca channels serve to damp depolarization and terminate the [Ca²⁺]_c increase after initial Ca²⁺ increase during the active phase.

Fig. 4.

Slow ATP/ADP ratio changes as a pacemaker. Burst behavior of the MP and the oscillation patterns of [Ca²⁺]_c, free ADP ([ADP]), and I_KATP are illustrated. It was simulated using a model (51) by a step increase of glucose level. Several variables, [Na⁺]_c, [IP₃], and [Ca²⁺]_ER, were frozen at constant levels to eliminate other mechanisms of fast and slow [Ca²⁺]_c oscillations ([Na⁺]_c = 7 mM, [IP₃] = 3 μM, and [Ca²⁺]_ER = 20 μM). Several coefficients were set as g_mSOC = 2 pS⁻¹ mV, g_mKATP = 200,000 pS, g_mKCa = 80 pS, g_mVCa = 710 pS, g_NaCa = 500 pS, k_ADP = 0.001 ms, k_ATP = 0.000001 ms, k_ATP,Ca = 0.0000218 μM/ms, and P_NaK = 400 fA. All other parameter settings are as in Ref. 51. A: action potential (V_P). B: [Ca²⁺]_c. C: [ADP]. D: I_KATP.

Negative [Ca²⁺]_c effects on ATP production.

In several mathematical models the uptake of Ca²⁺ by β-cell mitochondria has large energy-dissipative effects, leading to decreased oxidative phosphorylation suppressing the production rate of ATP (15, 103–105). This proposal also leads to a decreased [ATP]/[ADP] cytoplasmic ratio with increased [Ca²⁺]_c, as in first case, and the models can simulate [Ca²⁺]_c oscillations through the cyclical changes in [ATP]/[ADP] ratio (103–105). However, recent experimental data convincingly shows that “the primary role of mitochondrial Ca²⁺ is the stimulation of oxidative phosphorylation” (24), suggesting that [Ca²⁺]_c effects on ATP production are stimulatory rather than inhibitory.

Glycolytic oscillations as a mechanism for cyclical changes in the [ATP]/[ADP] ratio.

A number of hypotheses have been proposed to explain the mechanism of [ATP]/[ADP] ratio oscillations through glycolytic oscillations. The fundamental proposal of this model is that Ca²⁺-independent glycolytic oscillations that produce [ATP]/[ADP] ratio oscillations are mediated by the positive feedback of fructose 1,6-bisphosphate. This is the product of the glycolytic enzyme phosphofructokinase (PFK), stimulating PFK activity and subsequently depleting substrate (83, 142). Mathematical models for β-cell oscillations have been constructed on the basis of this mechanism (17–19, 81, 145).

However, the direct evidence for glycolytic oscillations in β-cells such as measurements of glycolytic metabolite concentrations is lacking. Periodic variations in mitochondrial membrane potential, NADH, and respiration have been offered as evidence of glycolytic oscillations (18). However, these phenomena can also be a consequence of independent Ca²⁺ cytoplasmic oscillations (see below). The available evidence supports the idea that glycolytic processes are not a requirement at least for slow cytoplasmic Ca²⁺ oscillations. Slow [Ca²⁺]_c oscillations can be induced at low glucose levels using K⁺ and K_ATP channel blockers (such as TEA and sulfonylureas) in islets and β-cell lines (see Fig. 5 and Refs. 51 and 125). Slow [Ca²⁺]_c oscillations can be generated even in the absence of glucose using such mitochondrial substrates as leucine (64), α-ketoisocaproic acid (69, 106), and methyl pyruvate (69). Fast [Ca²⁺]_c oscillations were found in islets incubated solely in the presence of glyceraldehyde or α-ketoisocaproic acid (31). PFK itself is not required for these oscillations, since [Ca²⁺]_c oscillations remain normal after 95–98% suppression of PFK-M activity in mouse islets (121). These data demonstrate that an oscillatory capacity is preserved when glycolytic flux is small or absent and that the slow components of [Ca²⁺]_c oscillations cannot be fully explained by mechanisms incorporating glycolytic oscillations. However, we cannot entirely rule out that this mechanism could contribute to ultrafast (and possibly fast) [Ca²⁺]_c oscillations, although the metabolic variability is on the order of minutes. Further studies are needed to establish whether there is any role at all for glycolytic-based mechanisms in β-cells regulating Ca²⁺ or other oscillations as opposed to periodic changes in mitochondrial Ca²⁺ or membrane potential. On other hand, glycolytic processes can modulate oscillation behavior. For example, bursting and Ca²⁺ oscillations can change frequency and amplitude and even disappear with changes in glucose concentration (54).

Fig. 5.

Effect of tolbutamide and thapsigargin on slow Ca²⁺ oscillations in mouse islets. Mouse islets were loaded with fura-2, and relative [Ca²⁺]_c changes were recorded as the 340/380 nm fluorescence excitation ratio changes obtained with the indicator fura-2 during incubation in 2 mM glucose with 50 μM tolbutamide, with subsequent addition of 1 μM thapsigargin.

Limitations of K_ATP channel [ATP]/[ADP] ratio-dependent mechanisms of oscillations.

The [ATP]/[ADP] ratio drops when [Ca²⁺]_c rises and increases when [Ca²⁺]_c falls, indirectly supporting the proposal that [Ca²⁺]_c oscillations can therefore evoke oscillations in [ATP]/[ADP] ratio and I_KATP (2, 34). These data are in agreement with the hypothesis that the K_ATP channel conductance changes can generate and maintain oscillations.

We have also considered above the mechanism for ultrafast oscillations determined by changes in the conductance of slow-activated Ca²⁺-sensitive K⁺ channels. Based on the effects of tolbutamide, a specific K_ATP channel blocker, it was concluded that K_ATP channels contribute a significant fraction (48%) of this mechanism (86). Therefore, we suggest that the mechanism underlying ultrafast bursts and [Ca²⁺]_c oscillations in β-cells can be due partly to the changes in the [ATP]/[ADP] ratio as a pacemaker (and corresponding changes in K_ATP channel conductivity), in accord with the K_ATP channel-dependent mechanism.

However, there are several valid arguments against a K_ATP channel-dependent mechanism for slow bursting and [Ca²⁺]_c oscillations. If K_ATP channels function as a pacemaker, the conductance of the K_ATP channels should oscillate during bursting electrical activity following changes in the [ATP]/[ADP] ratio. However, effective K_ATP channel blockers such as sulfonylureas do not stop slow bursting and Ca²⁺ oscillations in pancreatic islets and can even induce them in single β-cells or β-cell clusters isolated from mouse islets (see Fig. 5 and Refs. 43, 65, and 109) and in βTC3-neo cells (125). Slow [Ca²⁺]_c oscillations were also found in a mouse lacking functional K_ATP channels (40, 67, 120, 135). These results suggest that changes in the [ATP]/[ADP] ratio and, consequently, in K_ATP channel conductivity are unlikely to function as a pacemaker component for coupled slow oscillatory mechanisms.

However, studies are needed to carefully examine whether fast Ca²⁺ oscillations persist upon blocking K_ATP channels at low glucose levels or in K_ATP channel-null mice. Until such studies are reported, it is possible that ultrafast and fast bursting and the corresponding [Ca²⁺]_c oscillations are in part due to [ATP]/[ADP] ratio and K_ATP channel conductance changes acting as a pacemaker.

Intracellular [Na⁺] As a Slow Component in Pacemaker Mechanisms

Several components regulate Na⁺ dynamics in β-cells. The most important is the electrogenic Na⁺,K⁺-ATPase that extrudes three Na⁺ ions in exchange for two K⁺ ions for each molecule of ATP hydrolyzed, generating a net outward flow of cations through the PM. This pump is highly active, since it determines the high cytoplasmic K⁺ concentration (113). Like most other cells, the β-cells are also equipped with a PM Na⁺/Ca²⁺ exchanger, an electrogenic transporter that couples the exchange of 3 Na⁺ for 1 Ca²⁺ (52, 73). Consequently, changes in the cytoplasmic Na⁺ concentration ([Na⁺]_c) lead to changes of inward and outward currents and the PM potential. Changes in [Na⁺]_c can have a significant effect on generation of slow cytoplasmic Ca²⁺ oscillations in β-cells (56, 93).

For example, Kitasato et al. (93) proposed a specific oscillatory mechanism, including Na⁺ pumps and transporters. They suggested that increased [Ca²⁺]_c during an active phase results in an activation of the Na⁺/Ca²⁺ exchanger and a corresponding increase in [Na⁺]_c. An increase in cytosolic Na⁺ would lead to an enhanced ATP consumption at Na⁺/K⁺ pump sites and to a decreased ATP/ADP ratio. This in turn could lead to activation of K_ATP channels followed by membrane repolarization. A decrease in [Ca²⁺]_c results in the opposite order of events during a silent phase. This hypothesis was modeled by Miwa and Imai (110) as the underlying mechanism of [Ca²⁺]_c oscillations. This mechanism is a variant of K_ATP channel-dependent mechanism for a generation of bursts and [Ca²⁺]_c oscillations. It does not explain the experimental data, suggesting that the [ATP]/[ADP] ratio and K_ATP channel conductance changes are not the main driving force for slow [Ca²⁺]_c oscillations (see above).

We proposed and modeled a mechanism with a Ca²⁺ feedback effect where the intracellular [Na⁺]_c is a dynamic pacemaker variable that can govern bursts and slow [Ca²⁺]_c oscillations, although K_ATP channels do not change their activity (51). Figure 6 illustrates the roles of the Na⁺,K⁺-ATPase, the Na⁺/Ca²⁺ exchanger, and [Na⁺]_c dynamics in this model. Other slow parameters such as cytoplasmic concentrations of [ADP], [Ca²⁺]_ER, and [IP₃] were frozen at constant levels, blocking other possible mechanisms of bursting and [Ca²⁺]_c oscillations. The following mechanism was responsible for bursting and Ca²⁺ oscillations; increased [Ca²⁺]_c during an active period activates Na⁺ influx through Na⁺/Ca²⁺ exchangers. The resultant increase in intracellular [Na⁺]_c leads to a slow increase in outward current through electrogenic Na⁺/K⁺ pumps (I_NaK in Fig. 1), with corresponding plasma membrane repolarization, so that the cell reenters a silent phase. Decreased [Ca²⁺]_c during a silent phase leads to a slow decrease in [Na⁺]_c because Na⁺ influx through the Na⁺/Ca²⁺ exchanger decreases. This leads to a decreased outward current through Na⁺/K⁺ pumps and then in turn to PM depolarization and a generation of bursts and [Ca²⁺]_c oscillations.

Fig. 6.

Slow [Na⁺]_c changes as a pacemaker. Typical computer simulations using the model by Fridlyand et al. (51). Slow bursting and the oscillation patterns of Ca²⁺ and Na⁺ are illustrated. The other slow variables were frozen ([ATP] = 3,300 μM, [IP₃] = 3 μM, and [Ca²⁺]_ER = 50 μM) at constant levels to eliminate other mechanism of [Ca²⁺]_c oscillations. All other parameter settings are as in Ref. 51. A: V_P. B: [Ca²⁺]_c. C: [Na⁺]_c. D: I_NaK.

The effect of modulators leads to changes in the shape of simulated [Ca²⁺]_c oscillations, bringing them closer to experimental results. For example, Fig. 7 [which reproduces the third figure from our earlier article (51)] shows slow-simulated [Ca²⁺]_c oscillations that are similar to the experimental data represented in Fig. 2B. In this simulation, as opposed to that in Fig. 6, [ADP], [Ca²⁺]_ER, and [IP₃] were not frozen at constant levels and thus played a role as modulators. In this case, cytoplasmic [Na⁺] as a slowly variable component functioned as the pacemaker mechanism. We will discuss the possible mechanisms of this kind of influence of modulators on the shape of Ca²⁺ oscillators below.

Fig. 7.

Simulation of glucose-induced slow oscillations and SERCA blocking. Glucose-induced slow electrical bursting and [Ca²⁺]_c oscillations were simulated at a step increase of the rate constant of ATP production from a low to an intermediate value at t = 0. A: [Ca²⁺]_c. B: MP. C: Ca²⁺ in ER. D: cytosolic ATP. E: cytosolic IP₃. F: intracellular Na⁺ concentration. For simulation of thapsigargin action, the maximal rate of ER (SERCA; P_CaER) was decreased from 0.105 to 0.013 μM/ms at arrow in B (adapted from Ref. 51).

Some indirect experimental evidence supports the hypothesis that cyclic [Na⁺]_c changes can act as a pacemaker to induce bursting and slow [Ca²⁺]_c oscillations. For example, in isolated individual β-cells or cell clusters, the slow glucose-induced Na⁺ oscillations were found under conditions known to induce oscillations of [Ca²⁺]_c. Partial suppression of the Na⁺/K⁺ pump by ouabain resulted in an increased amplitude of slow Na⁺ and Ca²⁺ oscillations with decreased frequency; slow Na⁺ and Ca²⁺ oscillations were converted to a tonically elevated Na⁺ and Ca²⁺ level following ouabain inhibition of the Na,K-ATPase (62, 63). Interestingly, slow Ca²⁺ oscillations were converted to a sustained elevated Ca²⁺ level following ouabain even in sulfonylurea type 1 receptor (SUR1)-null mouse islets, where K_ATP channels were blocked (39).

Our computational simulation resulted in a similar picture (see the 7th figure in Ref. 51) showing that these experiments can be modeled exactly by a partial inhibition of the Na⁺/K⁺ pumps by oubain. According to this simulation, decreasing the Na⁺/K⁺ pump maximal activity reduced the rate of Na⁺ efflux and corresponding net outward current, leading to increased [Na⁺]_c and PM depolarization. This resulted in increased amplitude of slow Na⁺ and Ca²⁺ oscillations with decreased frequency. Greater inhibition of the Na⁺/K⁺ pump maximal activity led to a collapse of this Na⁺-dependent mechanism, the disappearance of bursting and slow Ca²⁺ oscillations, and tonically elevated [Ca²⁺]_c.

Veratridine causes voltage-dependent Na⁺ channels to stay open during a sustained membrane depolarization by decreasing inactivation (143). Partial activation of Na⁺ channels by veratridine resulted in a decreased amplitude and increased frequency of slow Ca²⁺ oscillations, and slow Ca²⁺ oscillations were converted to a tonically elevated Ca²⁺ level in isolated individual β-cells or cell clusters (44). We were able obtain similar results in isolated pancreatic islets (Fig. 8).

Fig. 8.

Effect of veratridine on slow Ca²⁺ oscillations in mouse islets. The experiments were performed in medium containing 2.5 mM Ca²⁺ and 14 mM glucose. Veratridine (μM) was added as indicated. The concentration of cytoplasmic Ca²⁺ is presented as the 340/380 nm fluorescence excitation ratio obtained with the indicator fura-2.

We simulated the effects of veratridine by shifting the half-activation potential for Na⁺ channels toward a more positive voltage using our general model (51). According to this simulation (Fig. 9), increased Na⁺ influx through voltage-dependent Na⁺ channels leads to an additional PM depolarization as a result of increased Na⁺ inward current and an accelerated Na⁺ accumulation during an active phase. This resulted in increased amplitude and frequency of slow Na⁺ and Ca²⁺ oscillations. The simulations are in reasonably good agreement with experimental data (compare Figs. 8 and 9).

Fig. 9.

Simulated influence of veratridine on slow [Ca²⁺]_c oscillations in β-cells. Addition of veratridine was simulated by decrease of half-activated potential for Na⁺ channels in the model by Fridlyand et al. (51). (Half-activated potential for Na⁺ channels was decreased from −104 to −94 mV at arrow 1 and from −94 to −84 mV at arrow 2 in B). A: V_P. B: [Ca²⁺]_c. C: [Na⁺]_c. D: I_Na.

Correlating the model simulations with experimental data shows that cyclic [Na⁺]_c changes can act as a pacemaker to induce bursting and slow [Ca²⁺]_c oscillations as opposed to the proposals on the role of K_ATP channels. However, this mechanism underlying oscillations remains incompletely investigated. Simultaneous measurements of [Ca²⁺]_c and [Na⁺]_c, although difficult, can directly test this hypothesis.

Role of the ER and [Ca²⁺]_ER as a Slow Pacemaker

The ER is a high-affinity and high-capacity organelle for calcium storage. The β-cell ER sequesters Ca²⁺ when the cytoplasmic calcium level is high and releases it when [Ca²⁺]_c is low or under the influence of intracellular messengers (55, 122, 123, 141). Ca²⁺ from the cytosol is transported into the ER by P-type ATPases using ATP and exits through two types of ER Ca²⁺ channels, the IP₃ receptor (IP₃R) channels and the ryanodine receptor channels (Fig. 1) (56, 82, 123, 138). However, Ca²⁺ influx from the extracellular space through VDCCs is the principal regulator of cytoplasmic Ca²⁺ homeostasis in β-cells under physiological conditions (6, 51, 127, 146).

[Ca²⁺]_ER and store operating currents.

Changes in [Ca²⁺]_ER may regulate PM potential. Depletion of intracellular Ca²⁺ stores in β-cells activates a special Ca²⁺ release-activated current [or store operating current (SOC)] through nonselective PM cation channels and can represent an inward cation current, leading to PM depolarization that potentiates glucose-induced Ca²⁺ influx trough VDCCs (20, 30, 97, 125, 147). This current may be Na⁺ influx through nonselective PM cation channels, which have been described in insulin-secreting β-cells (30, 97, 125). Transient receptor potential (TRP) ion channels were proposed for this role (74, 78, 125). Although the detailed mechanism for coupling of ER Ca²⁺ store depletion with the nonselective PM channels activation is uncertain, it likely involves translocation of STIM1 and STIM2 proteins from the ER calcium stores to the PM, where they interact with ORAI1 and related proteins associated with cation influx channels (119, 137).

Several mechanisms of oscillations with [Ca²⁺]_ER as a pacemaker employing a Ca²⁺ feedback effect have been proposed (20, 25, 51, 53, 107, 125, 148, 150). These proposals were primarily based on the characteristic properties of SOC that lead to cell depolarization with decreased [Ca²⁺]_ER. Several mathematical models have also been developed for these proposals using empirical equations for activation of SOC with decreased [Ca²⁺]_ER (20, 25, 53, 150). SOC increased with decreased [Ca²⁺]_ER and vice versa in simulations using the empirical equations from these models, yielding periodical changes of [Ca²⁺]_ER, bursting, and [Ca²⁺]_c oscillations.

We developed a model where “SOC” includes Na⁺ influx through nonselective cation channels that open following decreases in [Ca²⁺]_ER (SOC was termed “CRAN current” in our model) (51), following the earlier work of Roe et al. (125).

The proposed mechanism is illustrated in Fig. 10. In this case, other possible pacemaker mechanisms of [Ca²⁺]_c oscillations in the model (51) were blocked by freezing the cytoplasmic concentrations of [ADP], [Na⁺], and [IP₃] at constant levels. In this simulation, a rapid rise in [Ca²⁺]_c at the beginning of the active phase led to increased Ca²⁺ pumping into the ER and slow [Ca²⁺]_ER accumulation with corresponding closure of PM nonselective cation channels. This decreased inward Na⁺ influx current (I_SOC; Fig. 1) through these channels (i.e., SOC) resulted in PM repolarization, termination of the spiking, and the transition to a silent phase. Slowly decreasing [Ca²⁺]_ER during a silent phase as a result of Ca²⁺ leak through ryanodine or IP₃R then led to increased nonselective cation channel conductance, increased inward Na⁺ current, and PM depolarization. This resulted in subsequent activation of VDCCs and a new burst. In this case, [Ca²⁺]_ER is a slow pacemaker component for [Ca²⁺]_c oscillations with a Ca²⁺ feedback effect.

Fig. 10.

Slow [Ca²⁺]_ER changes as a pacemaker. Simulations were made as in Fig. 4. [Ca²⁺]_ER is the only slow parameter at frozen ADP, ATP, Na⁺, and IP₃ concentrations that eliminated other possible mechanisms of bursting and Ca²⁺ oscillations in the model by Fridlyand et al. (51). ([Na⁺]_c = 6 mM, [IP₃] = 1 μM, [ATP] = 3,400 μM, and k_ADP = 0.00045 ms⁻¹). All other parameter settings are as in Fridlyand et al. (51). A: V_P. B: [Ca²⁺]_c. C: [Ca²⁺]_ER. D: I_SOC.

However, an explanation of slow oscillations on the basis of [Ca²⁺]_ER-dependent mechanisms involves several difficulties, since it seems to be at odds with data demonstrating that the slow oscillations can persist in the presence of thapsigargin, a SERCA inhibitor (Fig. 5 and Refs. 6, 51, 55, 98, 109, and 136). Slow [Ca²⁺]_c oscillations also persist in islets from a SERCA3 knockout mouse, where fast oscillations disappear (11). These results argue against a role of [Ca²⁺]_ER changes as a pacemaker for slow [Ca²⁺]_c oscillations.

Slow [Ca²⁺]_c oscillations can also persist even though K_ATP channels and SERCA activity are blocked simultaneously in mouse islets by tolbutamide and thapsigargin (Fig. 5) or in isolated islets from the SUR1-null mouse lacking K_ATP channels when SERCA is inhibited (39). These complex experimental data show that a mechanism of slow [Ca²⁺]_c oscillations cannot be explained solely by an interaction of K_ATP channels and ER-dependent mechanisms.

SERCA blockade has led to conflicting results for fast bursts and [Ca²⁺]_c oscillations in β-cells converted to a sustained elevated [Ca²⁺]_c level by thapsigargin (55, 86, 109, 136, 147). However, Tamarina et al. (138) found that an addition of thapsigargin was accompanied by a gradual increase in the amplitude and frequency of fast [Ca²⁺]_c oscillations in some islets. Bertram et al. (20) had previously found that thapsigargin induced a gradual increase in the amplitude plateau fraction and in electrical burst frequency in the β-cell membrane potential. It is possible that in some experiments only partial SERCA blockade was obtained, leading to a partial decrease in [Ca²⁺]_ER, thereby preserving the ability of the β-cells to create an oscillatory mechanism. More profound SERCA block in other experiments could decrease [Ca²⁺]_ER to a threshold level where oscillatory mechanisms with [Ca²⁺]_ER as a pacemaker are suppressed. This suggests that the mechanism discussed above in which [Ca²⁺]_ER is the pacemaker component could be responsible for fast bursting and [Ca²⁺]_c oscillations.

Recently, null mice for one of the TRP ion channels (Trpm5) were found to have islets that had lost the ability to create fast [Ca²⁺]_c oscillations, whereas slow [Ca²⁺]_c oscillations were intact (27). The disappearance of fast [Ca²⁺]_c oscillations in this case is consistent with the hypothesis that Trpm5 channels can be a part of the SOC (or CRAN) mechanism with Na⁺ influx (75, 118) that is in part responsible for the fast, not slow, [Ca²⁺]_c oscillation (see above).

[Ca²⁺]_ER changes can regulate the shape of slow glucose-induced [Ca²⁺]_c oscillation in pancreatic β-cells (11, 99). Our simulations also support the hypothesis that slow [Ca²⁺]_c oscillation frequency and the generation of the waveform can be modulated by changes in the rates of Ca²⁺ leak from the ER, although [Ca²⁺]_ER does not account for the pacemaker mechanism for slow oscillations per se (51, 138). For example, Fig. 7 (reproduced from Ref. 51) shows the changes in the shape of [Ca²⁺]_c oscillations during simulation of SERCA block that lead to a decrease in [Ca²⁺]_ER. This is at conditions where the changes in [Na⁺]_c drive the pacemaker mechanism for slow [Ca²⁺]_c oscillations. In this case, Ca²⁺ is gradually released into the cytosol, and a decrease in [Ca²⁺]_c can be slow at the beginning of the silent period of slow oscillations (Fig. 7). When [Ca²⁺]_ER is small after SERCA inhibition, no additional Ca²⁺ release occurs during the silent period of slow oscillations, and the [Ca²⁺]_c decrease is sharp. A similar result was obtained experimentally (51, 55).

Pacemaker due to [Ca²⁺]_ER regulated by IP₃.

In pancreatic β-cells, as well as in other cell types, external agonists can act on G protein-coupled receptors to stimulate phospholipase C (PLC) activity. Ca²⁺-activated PLC catalyzes the formation of IP₃ from phosphatidylinositol-4,5-bisphosphate (Fig. 1) (23). The resulting rise in IP₃ initiates Ca²⁺ discharge from the ER through IP₃R channels (13, 56, 138). IP₃ can also activate K⁺ or Na⁺ conductances that can lead to PM repolarization (5, 134).

Changes in [IP₃]_c can play a crucial role in creating Ca²⁺ oscillations through activation of the ER IP₃ receptor in nonelectrically excitable types of cells (13, 130). Several mathematical models employed this mechanism for nonelectrically excitable cells. The first step is that a [Ca²⁺]_c increase is caused by Ca²⁺ release from the ER. One intracellular signal for this release is IP₃, which triggers Ca²⁺ release by binding to the IP₃Rs on the surface of the ER. This opens ER Ca²⁺ channels, allowing Ca²⁺ to flow out into the cytoplasm. Subsequent inactivation of the IP₃Rs and removal of Ca²⁺ by ER membrane pumps then reduces the Ca²⁺ concentration to lower levels. It can initiate a repeating cycle of release and uptake of Ca²⁺ from the ER, resulting in Ca²⁺ oscillations if the IP₃ concentration is in the correct range (13, 130). However, β-cells are electrically excitable cells, and as discussed above the changes in PM potential and corresponding Ca²⁺ influx from the extracellular space through VDCCs may be the principal regulator of cytoplasmic Ca²⁺ homeostasis in β-cells. For this reason, [IP₃]_c and IP₃R interactions are not likely to function as a pacemaker for [Ca²⁺]_c oscillations in β-cells.

We recently investigated the dynamics of IP₃ in mouse primary β-cells and in MIN6 mouse insulinoma cells upon stimulation with glucose and a muscarinic agonist (carbamoylcholine) (138). We found that IP₃ concentrations in the cytoplasm oscillated in parallel with [Ca²⁺]_c oscillations upon stimulation with glucose. Although this could be interpreted as evidence supporting IP₃ oscillations as a pacemaker of Ca²⁺ oscillations, in this case IP₃ oscillations can be readily explained by Ca²⁺ activation of PLC leading to [IP₃]_c changes, as simulated in our model (Fig. 7E). This suggests that slow cyclical changes in IP₃ concentration may follow from the activation of PLC by independent cyclical [Ca²⁺]_c changes; i.e., IP₃ may not be a pacemaker for [Ca²⁺]_c oscillations (51, 138). However, this problem needs further investigation, perhaps by PLC inhibition, to determine the existence of [Ca²⁺]_c oscillations in the absence of IP₃ oscillations.

However, the rate of IP₃ production can have a dramatic effect on oscillations even though IP₃ is not a pacemaker compound; i.e., IP₃ can be a strong modulator of oscillatory processes. For example, activation of IP₃ synthesis can cause a transformation of slow oscillations into fast ones (20, 138). Although the mechanism is unclear, it may be due to additional PM depolarization following from the opening of SOC or the recently found NALCN channels (134) with decreased [Ca²⁺]_ER, or this could follow from activation of PM K⁺ channels by IP₃.

cAMP as a Pacemaker Component

The cAMP pathway has [Ca²⁺]_c-dependent components such as some isoforms of adenylyl cyclase and phosphodiesterases (PDEs) (Fig. 1) (29, 60). cAMP may also increase Ca²⁺-induced Ca²⁺ release from the ER through ryanodine and IP₃Rs in mouse pancreatic β-cells and may activate some PM channels via phosphorylation or protein binding (66, 84, 85, 131). The suggestion was made for several types of cells that [Ca²⁺]_c oscillations could result from interactions between cAMP and [Ca²⁺]_c (61, 102). Therefore, an oscillatory mechanism with Ca²⁺ feedback effect and [cAMP]_c as a slow pacemaker component could also occur in β-cells.

However, recent experimental data does not support the hypothesis that cAMP is a pacemaker component for an oscillatory mechanism, at least for slow [Ca²⁺]_c oscillations in β-cells. For example, in insulinoma β-cell lines a PDE inhibitor (IBMX) abolished cAMP oscillations without inhibiting [Ca²⁺]_c oscillations (see the 5th figure in Ref. 95). Dyachok et al. (42) showed that IBMX can stimulate [Ca²⁺]_c oscillations even during elevated, nonoscillating cAMP levels, measured using real-time imaging. For this reason, according to our rules for hypothesis rejection (see introduction), we believe that cAMP oscillations are not part of a pacemaker mechanism, at least for slow [Ca²⁺]_c oscillations. We recently analyzed oscillatory behavior of cAMP using our model of cAMP dynamics and found that slow [Ca²⁺]_c oscillations are a pacemaker for cAMP oscillations in pancreatic β-cells and not the other way around (47).

However, cAMP (as well as IP₃) may modulate the dynamics of the intracellular processes and PM potential, leading to the changes in the oscillation pattern. For example, both IBMX and forskolin, an activator of adenyl cyclase, increase the level of cAMP, transforming [Ca²⁺]_c oscillations from slow into fast and further increasing their frequency (10). Other agents that increase the concentration of cAMP, such as glucagon, also promote the appearance of fast [Ca²⁺]_c oscillations (99). The underlying mechanisms of these transformations remain an open question but can be partially explained by altering PM depolarization.

The Pacemaker Role of Metabolic Oscillations

Metabolic cycles can control glucose-stimulated insulin secretion (80), and a number of hypotheses have implicated metabolic oscillations in determining Ca²⁺ oscillations (101, 142). Oscillations of a metabolic signal in β-cells may result from intrinsic properties of various pathways, although glycolytic oscillations are not likely to account for the mechanism of other metabolic oscillations (see above). We cannot rule out that some Ca²⁺-independent mechanism can create oscillations of compounds that can change the PM potential, leading to [Ca²⁺]_c oscillations. For example, based on the existence of citrate oscillations in isolated islet mitochondria, it was proposed that there is an independent mitochondrial oscillator and that the exported citrate may feed back and coordinate mitochondrial and cytosolic oscillations (101). Several mathematical models of possible mitochondrial oscillatory mechanisms were discussed recently (14), and these should be testable in β-cells.

Oscillations of the metabolic signals in β-cells may be secondary to oscillations in [Ca²⁺]_c. As was pointed out by Kennedy et al. (89), an appearance of metabolic oscillations does not prove that metabolic variations are the real pacemakers of oscillations, because these metabolic changes can indeed reflect the [Ca²⁺]_c oscillations. For example, variations in [Ca²⁺]_c can change ATP consumption because the rate of work of ATP-consuming Ca²⁺ pumps depends on Ca²⁺ concentrations. Using our mathematical model for β-cell processes (49, 50), we found that cyclic [ATP] and [ADP] oscillations in the cytoplasm due to the oscillations in [Ca²⁺]_c can stimulate the oscillations in metabolic processes such as oxygen consumption, the mitochondrial inner membrane potential, the rate of ATP production in mitochondria, and NAD(P)H levels in β-cells, i.e., in the oscillating processes that were found in experiments using β-cells (89, 92, 100, 111). We can also expect oscillations in metabolism if specific metabolic enzymes can be modulated by cyclic Ca²⁺ concentration changes in cytoplasm or mitochondria. For this reason, the oscillations in metabolic variables mentioned above are most likely secondary and likely follow rather than precede bursting and [Ca²⁺]_c oscillations.

Activation or Deactivation of the Voltage-Dependent Channels As a Pacemaker Component

As we pointed out in the introduction, the cyclic activation and deactivation of several channels can underlie mechanisms that drive bursting and Ca²⁺ oscillations in β-cells. Here, we consider other possible suggestions related to regulation of ion channels.

For example, it was suggested that PM repolarization and termination of spikes could occur during a burst if some parameters of the voltage-gated Ca²⁺ current could be slowly changed, leading to a decreased current (28). On the other hand, a regenerative recovery from inactivation and reactivation of Ca²⁺ currents in the rest period could lead to a depolarization and a beginning of a new burst. This mechanism for burst generation requires mathematical modeling of time-dependent changes in maximal conductance or other coefficients in the equations for VDCC (88). However, this mechanism for the changes in Ca²⁺ current also lacks sufficient experimental support.

Similarly, we can suggest that slow activation of any K⁺ currents (other than current through K_ATP and slow K_Ca channels, which we have considered above) in the active period can lead to repolarization and burst termination. A regenerative deactivation of these currents in the silent interval can lead to depolarization and to a beginning of a burst. However, the available experimental evidence argues against this hypothesis, at least for slow [Ca²⁺]_c oscillations. For example, stimulation of βTC3-neo or MIN6 insulinoma cells with glucose results only in a modest nonoscillatory increase in [Ca²⁺]_c. However, subsequent addition of TEA (15–20 mM, sufficient to substantially suppress most non-K_ATP K⁺ channels) can induce fast and slow [Ca²⁺]_c oscillations (41, 136, 139) so that inhibiting most K⁺ channels activates rather than suppresses [Ca²⁺]_c oscillations. Slow bursting and [Ca²⁺]_c oscillations are not lost in mouse islets lacking K_V2.1 channels (77). According to our rule for rejection, this indicates that K_V2.1 channels are not a part of a pacemaker mechanism, but they do have important modulating effects.

Interactions of Mechanisms and Compound Patterns of Bursting and [Ca²⁺]_c Oscillations

Fast and slow modes of oscillation can occur together, as in the mixed pattern, or separately, as in the fast and slow patterns (Fig. 2; see also Refs. 19 and 31). This strongly argues that they stem from distinct mechanisms. These mechanisms can be linked reciprocally and often occur together and can also proceed largely independently of each other. Mixed oscillations also occur in isolated cells, suggesting that this peculiar pattern does not result from the sum of signals produced in distinct β-cells in islets (6).

Modeling approaches show that compound bursting and corresponding mixed [Ca²⁺]_c oscillations can be simulated. The results of the modeling favor a scenario where two or more slow variables interact to produce complex oscillations. For example, Bertram and colleagues (17, 18) simulated the ultrafast and fast oscillations by small variations in K_ATP conductance using the empirical equation connecting the [ATP]/[ADP] ratio with [Ca²⁺]_c changes. The slow [Ca²⁺]_c oscillations were simulated as metabolic glycolytic oscillations such that the two interact to produce compound oscillations consisting of episodes of bursts separated by long periods of silence, or “accordion” oscillations, that consist of fast bursts with a slowly modulated duty cycle. Similar behavior could be achieved by defining the different slow and fast pacemaker variables (16). For example, slow [Ca²⁺]_c oscillations in a compound picture may be a result of a [Ca²⁺]_ER-dependent mechanism, and a K_ATP-dependent mechanism may be responsible for fast [Ca²⁺]_c oscillations on the top of the slow (35). According to our analysis, it is most likely that [Na⁺]_c can represent a slow-changing compound for slow oscillations and changes in the [ATP/ADP] ratio and/or [Ca²⁺]_ER for fast oscillations.

Modulators of Oscillation Behavior and External Pacemaker Mechanisms

Changes in channel conductance or Ca²⁺ handling can change PM polarization or Ca²⁺ redistribution. This, in turn, may lead to changes in bursting and [Ca²⁺]_c oscillations even though the cellular components do not participate directly in pacemaker oscillatory mechanisms. For example, activation of anion channels may cause depolarization of the β-cell PM (21, 22). Glucose stimulation of β-cells is coupled to an increase in Cl⁻ permeability, and oscillatory Ca²⁺ signaling may be critically dependent on transmembrane Cl⁻ fluxes (45). This indicates that agents that regulate anion channels may modulate oscillations. Cyclic nucleotide-gated (HCN) channels may also have a modulating effect because they can influence β-cell PM potential (152). External insulin can also be considered a modulator because it may be able to activate K_ATP channels (90).

Cyclical changes in external modulators can create or synchronize the alterations in PM polarization as well as the corresponding intracellular Ca²⁺ concentration. This property of β-cells can be important for synchronization of oscillations of insulin release in vivo in the pancreas. This synchronization is likely achieved through intrapancreatic innervation. This is probably due to cholinergie effects synchronizing islets (1, 149). Interestingly, a single cholinergie pulse per several internal [Ca²⁺]_c oscillations can synchronize these oscillations (46, 149).

Other external pacemaker mechanisms can also contribute to islet synchronization. For example, insulin secreted by islets acts on the liver, resulting in a reduction in plasma glucose concentration that can lead to glucose oscillations in blood, synchronizing an insulin secretion in islets (57, 114, 117). Additional synchronizing factors can be external ATP acting on purinergic receptors (115, 128).

Conclusion

Our analysis of several different hypotheses for pacemakers of oscillatory processes in β-cells shows that the following proposals may be most consistent with the existing experimental data. 1) Changes in the conductivity of slow K_Ca channels and K_ATP channels can be the pacemakers for ultrafast [Ca²⁺]_c oscillations; 2) changes in [ATP]/[ADP] ratio and/or [Ca²⁺]_ER can be the pacemakers for fast bursts and [Ca²⁺]_c oscillations; and 3) cyclical [Na⁺]_c changes may be important to explain the pattern of slow bursting and calcium oscillations. However, the exact physiological variables and their underlying mechanisms that drive bursting or Ca²⁺ oscillations in β-cells remain uncertain. Additional dynamic experiments and mathematical models are necessary to better understand this emerging aspect of β-cell physiology.

GRANTS

This work has been partially supported by National Institute of Diabetes and Digestive and Kidney Diseases Grants DRTC P60-DK-020595, DK-48494, and DK-063493 and a Research Grant from the Keck Foundation.

DISCLOSURES

The authors have nothing to disclose.

APPENDIX

Modeling of Slow-Activated Ca²⁺-Sensitive K⁺ Channel-Dependent Bursting and [Ca²⁺]_c Oscillations

Equations and coefficients for channels and pumps are taken from the previous model (51) or represented here. Slow-activated Ca²⁺-sensitive K⁺ channels as well as the fast-activated Ca²⁺-sensitive K⁺ channels were modeled, with considerably increased time-dependent parameters for activation reported in Ref. 59.

The units are time in milliseconds (ms), voltage in millivolts (mV), concentration in micromoles/liter (μM), current in femtoamperes (fA), conductance in picosiemens (pS), and capacitance in femtofarads (fF). Numerical integration of the model was carried out using standard numerical methods (51). This model is available for direct simulation on the web site “Virtual Cell” (www.nrcam.uchc.edu) in the “MathModel Database” on the “math workspace” in the library “Fridlyand” with the name “SlowCa+acted Kchan.”

Ca²⁺ Current (I_VCa)

$${\mathrm{I}}_{\text{VCa}}={\mathrm{g}}_{\text{mVCa}}{\mathrm{d}}_{\text{Ca}}{\mathrm{f}}_{\text{Ca}}({\mathrm{V}}_{\mathrm{P}}-{\mathrm{E}}_{\text{Ca}})$$ (1)

$$\frac{{\text{d d}}_{\text{Ca}}}{\text{dt}}=\frac{{\mathrm{d}}_{\text{Cab}}-{\mathrm{d}}_{\text{Ca}}}{{\tau }_{\text{dCa}}}$$ (2)

$${\mathrm{d}}_{\text{Cab}}=\frac{1}{1+\text{exp}\left[\right(2-{\mathrm{V}}_{\mathrm{P}})/10.5]}$$ (3)

$${\tau }_{\text{dCa}}=2.2-1.79\times \text{exp}-{[({\mathrm{V}}_{\mathrm{p}}-9.7)/70.2]}^2$$ (4)

$${\mathrm{f}}_{\text{VCa}}={\mathrm{f}}_{\text{VCab}}=\frac{1}{{1+\text{exp}\left[\right(-15+\mathrm{V}}_{\mathrm{P}})/6]},$$ (5)

where g_mVCa = 800 pS, E_Ca = 100 mV.

Delayed Rectifier K⁺ Current (I_Kr)

$${\mathrm{I}}_{\text{Kr}}={\mathrm{g}}_{\text{mKr}}{\mathrm{d}}_{\text{Kr}}^2{\mathrm{f}}_{\text{Kr}}{(\mathrm{V}}_{\mathrm{p}}-{\mathrm{E}}_{\mathrm{K}}),$$ (6)

$$\frac{{\text{d d}}_{\text{Kr}}}{\text{dt}}=\frac{{\mathrm{d}}_{\text{Krb}}-{\mathrm{d}}_{\text{Kr}}}{{\tau }_{\text{dKr}}}$$ (7)

$${\mathrm{d}}_{\text{Krb}\infty }=\frac{1}{1+\text{exp}\left[\right(-{9-\mathrm{V}}_{\mathrm{P}})/5]},$$ (8)

where g_mKr = 30,000 pS, τ_dKr = 23 mc, E_K = −75 mV.

Fast Ca²⁺-Activated K⁺ Current (I_KCa)

$${\mathrm{I}}_{\text{KCa}}={\mathrm{g}}_{\text{mKCa}}{\mathrm{d}}_{\text{KCa}}{(\mathrm{V}}_{\mathrm{p}}-{\mathrm{E}}_{\mathrm{K}}),$$ (9)

$${\mathrm{d}}_{\text{KCa}}=\frac{{[{\text{Ca}}^{2+}]}_{\mathrm{c}}^4}{{[{\text{Ca}}^{2+}]}_{\mathrm{c}}^4+{(0.25)}^4},$$ (10)

where g_mKCa = 10 pS.

Slow-Activated Ca²⁺-Sensitive K⁺ Channels

$${\mathrm{I}}_{\text{KCas}}={\mathrm{g}}_{\text{mKCas}}{\mathrm{d}}_{\text{KCas}}{(\mathrm{V}}_{\mathrm{p}}-{\mathrm{E}}_{\mathrm{K}}),$$ (11)

$$\frac{{\text{d d}}_{\text{KCas}}}{\text{dt}}=\frac{{\mathrm{d}}_{\text{KCasb}}-{\mathrm{d}}_{\text{KCas}}}{{\tau }_{\text{dKCas}}}$$ (12)

$${\mathrm{d}}_{\text{KCab}}=\frac{{[{\text{Ca}}^{2+}]}_{\mathrm{c}}^4}{{[{\text{Ca}}^{2+}]}_{\mathrm{c}}^4+{(0.25)}^4},$$ (13)

where G_mKCas = 90 pS and τ_dKCas = 2,300 mc.

ATP-Sensitive K⁺ Channel Current (I_KATP)

$${\mathrm{I}}_{\text{KATP}}={\mathrm{g}}_{\text{mKATP}}{\mathrm{O}}_{\text{KATP}}{(\mathrm{V}}_{\mathrm{p}}-{\mathrm{E}}_{\mathrm{K}}),$$ (14)

where

$${\mathrm{O}}_{\text{KATP}}=\frac{0.08(1+2{{[\text{MgADP}}_{\mathrm{f}}]}_{\mathrm{i}}/17)+0.89{({{[\text{MgADP}}_{\mathrm{f}}]}_{\mathrm{i}}/17)}^2}{{(1+{{[\text{MgADP}}_{\mathrm{f}}]}_{\mathrm{i}}/17)}^2(1+0.45{{[\text{MgADP}}_{\mathrm{f}}]}_{\mathrm{i}}/26+{{[\text{ATP}}_{\text{free}}\text{]}}_{\mathrm{i}}/50)}$$ (15)

$${[{\mathrm{MgADP}}_{\mathrm{f}}]}_{\mathrm{i}}=0.55{[{\mathrm{ADP}}_{\mathrm{f}}]}_{\mathrm{i}},$$ (16)

where O_KATP is the fraction of open K_ATP channels, [MgADP_f]_i is the concentration of free Mg-bound ADP, G_mKATP = 30,000 pS, and [ADP_f]_i = 20 μM.

Plasma Membrane Ca²⁺ Pump Current (I_Cap)

$${\mathrm{I}}_{\text{Cap}}={\mathrm{P}}_{\text{mCap}}\frac{{[{\text{Ca}}^{2+}]}_{\text{ci}}^2}{{[{\text{Ca}}^{2+}]}_{\mathrm{c}}^2+{(0.2)}^2},$$ (17)

where P_mCap = 4,300 fA.

Na⁺ Background Current (I_Nab)

$${\mathrm{I}}_{\text{Nab}}={\mathrm{g}}_{\text{mNab}}{(\mathrm{V}}_{\mathrm{p}}-{\mathrm{E}}_{\text{Na}}),$$ (18)

where g_mNab = 12 pS and E_Na = −70 mV.

The differential equation describing time-dependent changes in the PM potential (V_p) is the current balance equation:

$$-{\mathrm{C}}_{\mathrm{m}}\frac{{\text{dV}}_{\mathrm{p}}}{\text{dt}}={\mathrm{I}}_{\text{VCa}}+{\mathrm{I}}_{\text{Kr}}+{\mathrm{I}}_{\text{KATP}}+{\mathrm{I}}_{\text{KCa}}+{\mathrm{I}}_{\text{KCas}}+{\mathrm{I}}_{\text{Cap}}+{\mathrm{I}}_{\text{Nab}},$$ (19)

where C_m is the whole cell membrance capacitance. The PM currents are listed in Fig. 1.

Based on the consideration in our general model (51), and including only I_VCA and I_Cap, the equations for [Ca²⁺] dynamics can be written as

$$\frac{{\mathrm{d}[\text{Ca}}^{2+}]\mathrm{c}}{\text{dt}}=\frac{{\mathrm{f}}_{\mathrm{i}}(-{\mathrm{I}}_{\text{VCa}}-{2\mathrm{I}}_{\text{Cap}})}{2F{\mathrm{V}}_{\mathrm{i}}}-{\mathrm{k}}_{\text{sg}}{[\text{Ca}}^{2+}]\mathrm{c}$$ (20)

where f_i is the fraction of free Ca²⁺ in cytoplasm, F is Faraday's constant, V_i is the effective volumes of the cytosolic compartment, and k_sg is a coefficient of the sequestration rate of [Ca²⁺]_c.