The Ca2+ dynamics of isolated mouse beta-cells and islets: implications for mathematical models

[Ca(2+)](i) and electrical activity were compared in isolated beta-cells and islets using standard techniques. In islets, raising glucose caused a decrease in [Ca(2+)](i) followed by a plateau and then fast (2-3 min(-1)), slow (0.2-0.8 min(-1)), or a mixture of fast and slow [Ca(2+)](i) oscillations. In beta-cells, glucose transiently decreased and then increased [Ca(2+)](i), but no islet-like oscillations occurred. Simultaneous recordings of [Ca(2+)](i) and electrical activity suggested that differences in [Ca(2+)](i) signaling are due to differences in islet versus beta-cell electrical activity. Whereas islets exhibited bursts of spikes on medium/slow plateaus, isolated beta-cells were depolarized and exhibited spiking, fast-bursting, or spikeless plateaus. These electrical patterns in turn produced distinct [Ca(2+)](i) patterns. Thus, although isolated beta-cells display several key features of islets, their oscillations were faster and more irregular. beta-cells could display islet-like [Ca(2+)](i) oscillations if their electrical activity was converted to a slower islet-like pattern using dynamic clamp. Islet and beta-cell [Ca(2+)](i) changes followed membrane potential, suggesting that electrical activity is mainly responsible for the [Ca(2+)] dynamics of beta-cells and islets. A recent model consisting of two slow feedback processes and passive endoplasmic reticulum Ca(2+) release was able to account for islet [Ca(2+)](i) responses to glucose, islet oscillations, and conversion of single cell to islet-like [Ca(2+)](i) oscillations. With minimal parameter variation, the model could also account for the diverse behaviors of isolated beta-cells, suggesting that these behaviors reflect natural cell heterogeneity. These results support our recent model and point to the important role of beta-cell electrical events in controlling [Ca(2+)](i) over diverse time scales in islets.     

Three roads to islet bursting: emergent oscillations in coupled phantom bursters

Glucose-induced membrane potential and Ca(2+) oscillations in isolated pancreatic beta-cells occur over a wide range of frequencies, from >6/min (fast) to <1/min (slow). However, cells within intact islets generally oscillate with periods of 10-60 s (medium). The phantom bursting concept addresses how beta-cells can generate such a wide range of frequencies. Here, we explore an updated phantom bursting model to determine how heterogeneity in a single parameter can explain both the broad frequency range observed in single cells and the rarity of medium oscillations. We then incorporate the single-cell model into an islet model with parameter heterogeneity. We show that strongly coupled islets must be composed of predominantly medium oscillating single cells or a mixture of fast and slow cells to robustly produce medium oscillations. Surprisingly, we find that this constraint does not hold for moderate coupling, and that robustly medium oscillating islets can arise from populations of single cells that are essentially all slow or all fast. Thus, with coupled phantom bursters, medium oscillating islets can be constructed out of cells that are either all fast, all slow, or a combination of the two.     

Theoretical studies on the electrical activity of pancreatic beta-cells as a function of glucose.

The electrical activity of pancreatic beta-cells, which has been closely correlated both with intracellular Ca2+ concentration and insulin release, is characterized by a biphasic response to glucose and bursts of spiking action potentials. Recent voltage clamp and single channel patch clamp experiments have identified several transmembrane ionic channels that may play key roles in the electrophysiological behavior of beta-cells. There is a hypothesis that Ca2+-activated K+ channels are responsible for both the resting potential during low glucose concentration and the silent phase during bursting. The discovery of the ATP-inactivated K+ channel raises the possibility that the current for this latter K+ channel may dominate the resting potential, while the Ca2+-activated K+ current dominates the silent phase potential between bursts. The recent discovery that Ca2+-activated K+ channels are pH sensitive raises an interesting possibility for the biphasic electrical response. In this paper, numerical methods are presented for evaluating these hypotheses against experimental evidence.     

Role of single-channel stochastic noise on bursting clusters of pancreatic beta-cells.

To study why pancreatic beta-cells prefer to burst as a multi-cellular complex, we have formulated a stochastic model for bursting clusters of excitable cells. Our model incorporated a delayed rectifier K+ channel, a fast voltage-gated Ca2+ channel, and a slow Cai-blockable Ca2+ channel. The fraction of ATP-sensitive K+ channels that may still be active in the bursting regime was included in the model as a leak current. We then developed an efficient method for simulating an ionic current component of an excitable cell that contains several thousands of channels opening simultaneously under unclamped voltage. Single channel open-close stochastic events were incorporated into the model by use of binomially distributed random numbers. Our simulations revealed that in an isolated beta-cell [Ca2+]i oscillates with a small amplitude about a low [Ca2+]i. However, in a large cluster of tightly coupled cells, stable bursts develop, and [Ca2+]i oscillates with a larger amplitude about a higher [Ca2+]i. This may explain why single beta-cells do not burst and also do not release insulin.     

How noise and coupling induce bursting action potentials in pancreatic {beta}-cells

Unlike isolated beta-cells, which usually produce continuous spikes or fast and irregular bursts, electrically coupled beta-cells are apt to exhibit robust bursting action potentials. We consider the noise induced by thermal fluctuations as well as that by channel-gating stochasticity and examine its effects on the action potential behavior of the beta-cell model. It is observed numerically that such noise in general helps single cells to produce a variety of electrical activities. In addition, we also probe coupling via gap junctions between neighboring cells, with heterogeneity induced by noise, to find that it enhances regular bursts.     

Pharmacological properties and functional role of Kslow current in mouse pancreatic beta-cells: SK channels contribute to Kslow tail current and modulate insulin secretion

The pharmacological properties of slow Ca(2+)-activated K(+) current (K(slow)) were investigated in mouse pancreatic beta-cells and islets to understand how K(slow) contributes to the control of islet bursting, [Ca(2+)](i) oscillations, and insulin secretion. K(slow) was insensitive to apamin or the K(ATP) channel inhibitor tolbutamide, but UCL 1684, a potent and selective nonpeptide SK channel blocker reduced the amplitude of K(slow) tail current in voltage-clamped mouse beta-cells. K(slow) was also selectively and reversibly inhibited by the class III antiarrythmic agent azimilide (AZ). In isolated beta-cells or islets, pharmacologic inhibition of K(slow) by UCL 1684 or AZ depolarized beta-cell silent phase potential, increased action potential firing, raised [Ca(2+)](i), and enhanced glucose-dependent insulin secretion. AZ inhibition of K(slow) also supported mediation by SK, rather than cardiac-like slow delayed rectifier channels since bath application of AZ to HEK 293 cells expressing SK3 cDNA reduced SK current. Further, AZ-sensitive K(slow) current was extant in beta-cells from KCNQ1 or KCNE1 null mice lacking cardiac slow delayed rectifier currents. These results strongly support a functional role for SK channel-mediated K(slow) current in beta-cells, and suggest that drugs that target SK channels may represent a new approach for increasing glucose-dependent insulin secretion. The apamin insensitivity of beta-cell SK current suggests that beta-cells express a unique SK splice variant or a novel heteromultimer consisting of different SK subunits.     

Diffusion of extracellular K+ can synchronize bursting oscillations in a model islet of Langerhans.

Electrical bursting oscillations of mammalian pancreatic beta-cells are synchronous among cells within an islet. While electrical coupling among cells via gap junctions has been demonstrated, its extent and topology are unclear. The beta-cells also share an extracellular compartment in which oscillations of K+ concentration have been measured (Perez-Armendariz and Atwater, 1985). These oscillations (1-2 mM) are synchronous with the burst pattern, and apparently are caused by the oscillating voltage-dependent membrane currents: Extracellular K+ concentration (Ke) rises during the depolarized active (spiking) phase and falls during the hyperpolarized silent phase. Because raising Ke depolarizes the cell membrane by increasing the potassium reversal potential (VK), any cell in the active phase should recruit nonspiking cells into the active phase. The opposite is predicted for the silent phase. This positive feedback system might couple the cells' electrical activity and synchronize bursting. We have explored this possibility using a theoretical model for bursting of beta-cells (Sherman et al., 1988) and K+ diffusion in the extracellular space of an islet. Computer simulations demonstrate that the bursts synchronize very quickly (within one burst) without gap junctional coupling among the cells. The shape and amplitude of computed Ke oscillations resemble those seen in experiments for certain parameter ranges. The model cells synchronize with exterior cells leading, though incorporating heterogeneous cell properties can allow interior cells to lead. The model islet can also be forced to oscillate at both faster and slower frequencies using periodic pulses of higher K+ in the medium surrounding the islet. Phase plane analysis was used to understand the synchronization mechanism. The results of our model suggest that diffusion of extracellular K+ may contribute to coupling and synchronization of electrical oscillations in beta-cells within an islet.     

Carbon monoxide stimulates insulin release and propagates Ca2+ signals between pancreatic beta-cells

A key question for understanding the mechanisms of pulsatile insulin release is how the underlying beta-cell oscillations of the cytoplasmic Ca2+ concentration ([Ca2+]i) are synchronized within and among the islets in the pancreas. Nitric oxide has been proposed to coordinate the activity of the beta-cells by precipitating transients of [Ca2+]i. Comparing ob/ob mice and lean controls, we have now studied the action of carbon monoxide (CO), another neurotransmitter with stimulatory effects on cGMP production. A strong immunoreactivity for the CO-producing constitutive heme oxygenase (HO-2) was found in ganglionic cells located in the periphery of the islets and in almost all islet endocrine cells. Islets from ob/ob mice had sixfold higher generation of CO (1 nmol.min-1.mg protein-1) than the lean controls. This is 100-fold the rate for their constitutive production of NO. Moreover, islets from ob/ob mice showed a threefold increase in HO-2 expression and expressed inducible HO (HO-1). The presence of an excessive islet production of CO in the ob/ob mouse had its counterpart in a pronounced suppression of the glucose-stimulated insulin release from islets exposed to the HO inhibitor Zn-protoporhyrin (10 microM) and in a 16 times higher frequency of [Ca2+]i transients in their beta-cells. Hemin (0.1 and 1.0 microM), the natural substrate for HO, promoted the appearance of [Ca2+]i transients, and 10 microM of the HO inhibitors Zn-protoporphyrin and Cr-mesoporphyrin had a suppressive action both on the firing of transients and their synchronization. It is concluded that the increased islet production of CO contributes to the hyperinsulinemia in ob/ob mice. In addition to serving as a positive modulator of glucose-stimulated insulin release, CO acts as a messenger propagating Ca2+ signals with coordinating effects on the beta-cell rhythmicity.     

Glucose-induced oscillations of intracellular Ca2+ concentration resembling bursting electrical activity in single mouse islets of Langerhans

Intracellular Ca2+ levels were monitored in single, acutely isolated mouse islets of Langerhans by dual emission Indo-1 fluorometry. High-frequency (3.1 min-1) [Ca2+]i oscillations with a brief rising time (1-2 s) and 10 s half-width ('fast' oscillations) were detected in 11 mM glucose. Raising the glucose concentration to 16.7 mM increased the duration of these oscillations, which were otherwise absent in 5.5 mM glucose. [Ca2+]i waves of lower frequency (0.5 min-1) and longer rising time ('slow' oscillations) were also recorded. The data indicate that "fast" oscillations are directly related to beta-cell bursting electrical activity, and suggest the existence of extensive networks of electrically coupled cells in the islet.     

Phantom bursting is highly sensitive to noise and unlikely to account for slow bursting in beta-cells: considerations in favor of metabolically driven oscillations

Pancreatic beta-cells show bursting electrical activity with a wide range of burst periods ranging from a few seconds, often seen in isolated cells, over tens of seconds (medium bursting), usually observed in intact islets, to several minutes. The phantom burster model [Bertram, R., Previte, J., Sherman, A., Kinard, T.A., Satin, L.S., 2000. The phantom burster model for pancreatic beta-cells. Biophys. J. 79, 2880-2892] provided a framework, which covered this span, and gave an explanation of how to obtain medium bursting combining two processes operating on different time scales. However, single cells are subjected to stochastic fluctuations in plasma membrane currents, which are likely to disturb the bursting mechanism and transform medium bursters into spikers or very fast bursters. We present a polynomial, minimal, phantom burster model and show that noise modifies the plateau fraction and lowers the burst period dramatically in phantom bursters. It is therefore unlikely that slow bursting in single cells is driven by the slow phantom bursting mechanism, but could instead be driven by oscillations in glycolysis, which we show are stable to random ion channel fluctuations. Moreover, so-called compound bursting can be converted to apparent slow bursting by noise, which could explain why compound bursting and mixed Ca(2+) oscillations are seen mainly in intact islets.     

Calcium and glycolysis mediate multiple bursting modes in pancreatic islets

Pancreatic islets of Langerhans produce bursts of electrical activity when exposed to stimulatory glucose levels. These bursts often have a regular repeating pattern, with a period of 10-60 s. In some cases, however, the bursts are episodic, clustered into bursts of bursts, which we call compound bursting. Consistent with this are recordings of free Ca2+ concentration, oxygen consumption, mitochondrial membrane potential, and intraislet glucose levels that exhibit very slow oscillations, with faster oscillations superimposed. We describe a new mathematical model of the pancreatic beta-cell that can account for these multimodal patterns. The model includes the feedback of cytosolic Ca2+ onto ion channels that can account for bursting, and a metabolic subsystem that is capable of producing slow oscillations driven by oscillations in glycolysis. This slow rhythm is responsible for the slow mode of compound bursting in the model. We also show that it is possible for glycolytic oscillations alone to drive a very slow form of bursting, which we call "glycolytic bursting." Finally, the model predicts that there is bistability between stationary and oscillatory glycolysis for a range of parameter values. We provide experimental support for this model prediction. Overall, the model can account for a diversity of islet behaviors described in the literature over the past 20 years.     

Glucose induces temperature-dependent oscillations of cytoplasmic Ca2+ in single pancreatic beta-cells related to their electrical activity

Glucose induces large amplitude oscillations of the cytoplasmic Ca2+ concentration ([Ca2+]i) in pancreatic beta-cells. The effects of temperature on these oscillations were examined by monitoring [Ca2+]i continuously in single beta-cells from ob/ob-mice using dual wavelength microfluorometry. The oscillations of [Ca2+]i disappeared when the temperature was increased above 42 degrees C and were reversibly inhibited below 30 degrees C. However, cooling did not prevent a glucose response in terms of the average rise of [Ca2+]i. Since patch clamp studies of single beta-cells have indicated a random occurrence of glucose-induced action potentials at room temperature, it was important to explore how the sugar affected the electrical activity at 37 degrees C. Using the cell-attached configuration of the patch clamp technique for such analyses, the action potentials were found to occur in bursts with durations similar to the large amplitude oscillations of [Ca2+]i.     

The phantom burster model for pancreatic beta-cells

Pancreatic beta-cells exhibit bursting oscillations with a wide range of periods. Whereas periods in isolated cells are generally either a few seconds or a few minutes, in intact islets of Langerhans they are intermediate (10-60 s). We develop a mathematical model for beta-cell electrical activity capable of generating this wide range of bursting oscillations. Unlike previous models, bursting is driven by the interaction of two slow processes, one with a relatively small time constant (1-5 s) and the other with a much larger time constant (1-2 min). Bursting on the intermediate time scale is generated without need for a slow process having an intermediate time constant, hence phantom bursting. The model suggests that isolated cells exhibiting a fast pattern may nonetheless possess slower processes that can be brought out by injecting suitable exogenous currents. Guided by this, we devise an experimental protocol using the dynamic clamp technique that reliably elicits islet-like, medium period oscillations from isolated cells. Finally, we show that strong electrical coupling between a fast burster and a slow burster can produce synchronized medium bursting, suggesting that islets may be composed of cells that are intrinsically either fast or slow, with few or none that are intrinsically medium.     

Synchronization and entrainment of cytoplasmic Ca2+ oscillations in cell clusters prepared from single or multiple mouse pancreatic islets

In contrast to pancreatic islets, isolated beta-cells stimulated by glucose display irregular and asynchronous increases in cytoplasmic Ca(2+) concentration ([Ca(2+)](i)). Here, clusters of 5-30 cells were prepared from a single mouse islet or from pools of islets, loaded with fura-2, and studied with a camera-based system. [Ca(2+)](i) oscillations were compared in pairs of clusters by computing the difference in period and a synchronization index lambda. During perifusion with 12 mM glucose, the clusters exhibited regular [Ca(2+)](i) oscillations that were quasi-perfectly synchronized (Delta period of 1.4% and index lambda close to 1.0) between cells of each cluster. In contrast, separate clusters were not synchronized, even when prepared from one single islet. Pairs of clusters neighboring on the same coverslip were not better synchronized than pairs of clusters examined separately (distinct coverslips). We next attempted to synchronize clusters perifused with 12 mM glucose by applying external signals. A single pulse of 20 mM glucose, 10 mM amino acids, or 10 microM tolbutamide transiently altered [Ca(2+)](i) oscillations but did not reset the clusters to oscillate synchronously. On a background of 12 mM glucose, repetitive applications (1 min/5 min) of 10 microM tolbutamide, but not of 20 mM glucose, synchronized separate clusters. Our results identify a level of beta-cell heterogeneity intermediate between single beta-cells and the whole islet. They do not support the idea that substances released by islet cells serve as paracrine synchronizers. However, synchronization can be achieved by an external signal, if this signal has a sufficient strength to overwhelm the intrinsic rhythm of glucose-induced oscillations and is repetitively applied.     

Ionic mechanisms underlying spontaneous CA1 neuronal firing in Ca2+-free solution

Hippocampal CA1 neurons exposed to zero-[Ca(2+)] solutions can generate periodic spontaneous synchronized activity in the absence of synaptic function. Experiments using hippocampal slices showed that, after exposure to zero-[Ca(2+)](0) solution, CA1 pyramidal cells depolarized 5-10 mV and started firing spontaneous action potentials. Spontaneous single neuron activity appeared in singlets or was grouped into bursts of two or three action potentials. A 16-compartment, 23-variable cable model of a CA1 pyramidal neuron was developed to study mechanisms of spontaneous neuronal bursting in a calcium-free extracellular solution. In the model, five active currents (a fast sodium current, a persistent sodium current, an A-type transient potassium current, a delayed rectifier potassium current, and a muscarinic potassium current) are included in the somatic compartment. The model simulates the spontaneous bursting behavior of neurons in calcium-free solutions. The mechanisms underlying several aspects of bursting are studied, including the generation of triplet bursts, spike duration, burst termination, after-depolarization behavior, and the prolonged inactive period between bursts. We show that the small persistent sodium current can play a key role in spontaneous CA1 activity in zero-calcium solutions. In particular, it is necessary for the generation of an after-depolarizing potential and prolongs both individual bursts and the interburst interval.     

Delayed-rectifier (KV2.1) regulation of pancreatic beta-cell calcium responses to glucose: inhibitor specificity and modeling

The delayed-rectifier (voltage-activated) K(+) conductance (K(V)) in pancreatic islet beta-cells has been proposed to regulate plasma membrane repolarization during responses to glucose, thereby determining bursting and Ca(2+) oscillations. Here, we verified the expression of K(V)2.1 channel protein in mouse and human islets of Langerhans. We then probed the function of K(V)2.1 channels in islet glucose responses by comparing the effect of hanatoxin (HaTx), a specific blocker of K(V)2.1 channels, with a nonspecific K(+) channel blocker, tetraethylammonium (TEA). Application of HaTx (1 microM) blocked delayed-rectifier currents in mouse beta-cells, resulting in a 40-mV rightward shift in threshold of activation of the voltage-dependent outward current. In the presence of HaTx, there was negligible voltage-activated outward current below 0 mV, suggesting that K(V)2.1 channels form the predominant part of this current in the physiologically relevant range. We then employed HaTx to study the role of K(V)2.1 in the beta-cell Ca(2+) responses to elevated glucose in comparison with TEA. Only HaTx was able to induce slow intracellular Ca(2+) concentration ([Ca(2+)](i)) oscillations in cells stimulated with 20 mM glucose, whereas TEA induced an immediate rise in [Ca(2+)](i) followed by rapid oscillations. In human islets, HaTx acted in a similar fashion. The data were analyzed using a detailed mathematical model of ionic flux and Ca(2+) regulation in beta-cells. The results can be explained by a specific HaTx effect on the K(V) current, whereas TEA affects multiple K(+) conductances. The results underscore the importance of K(V)2.1 channel in repolarization of the pancreatic beta-cell plasma membrane and its role in regulating insulin secretion.     

Properties of voltage-gated Ca2+ currents measured from mouse pancreatic beta-cells in situ

We used the single-microelectrode voltage-clamp technique to record ionic currents from pancreatic beta-cells within intact mouse islets of Langerhans at 37 degrees C, the typical preparation for studies of glucose-induced "bursting" electrical activity. Cells were impaled with intracellular microelectrodes, and voltage pulses were applied in the presence of tetraethylammonium. Under these conditions, a voltage-dependent Ca2+ current (I(Cav)), containing L-type and non-L-type components, was observed. The current measured in situ was larger than that measured in single cells with whole-cell patch clamping, particularly at membrane potentials corresponding to the action potentials of beta-cell electrical activity. The temperature dependence of I(Cav) was not sufficient to account for the difference in size of the currents recorded with the two methods. During prolonged pulses, the voltage-dependent Ca2+ current measured in situ displayed both rapid and slow components of inactivation. The rapid component was Ca2+-dependent and was inhibited by the membrane-permeable Ca2+ chelator, BAPTA-AM. The effect of BAPTA-AM on beta-cell electrical activity then demonstrated that Ca2+-dependent inactivation of I(Cav) contributes to action potential repolarization and to control of burst frequency. Our results demonstrate the utility of voltage clamping beta-cells in situ for determining the roles of ion channels in electrical activity and insulin secretion.     

Investigating the role of islet cytoarchitecture in its oscillation using a new beta-cell cluster model

The oscillatory insulin release is fundamental to normal glycemic control. The basis of the oscillation is the intercellular coupling and bursting synchronization of beta cells in each islet. The functional role of islet beta cell mass organization with respect to its oscillatory bursting is not well understood. This is of special interest in view of the recent finding of islet cytoarchitectural differences between human and animal models. In this study we developed a new hexagonal closest packing (HCP) cell cluster model. The model captures more accurately the real islet cell organization than the simple cubic packing (SCP) cluster that is conventionally used. Using our new model we investigated the functional characteristics of beta-cell clusters, including the fraction of cells able to burst f(b), the synchronization index lambda of the bursting beta cells, the bursting period T(b), the plateau fraction p(f), and the amplitude of intracellular calcium oscillation [Ca]. We determined their dependence on cluster architectural parameters including number of cells n(beta), number of inter-beta cell couplings of each beta cell n(c), and the coupling strength g(c). We found that at low values of n(beta), n(c) and g(c), the oscillation regularity improves with their increasing values. This functional gain plateaus around their physiological values in real islets, at n(beta) approximately 100, n(c) approximately 6 and g(c) approximately 200 pS. In addition, normal beta-cell clusters are robust against significant perturbation to their architecture, including the presence of non-beta cells or dead beta cells. In clusters with n(beta)> approximately 100, coordinated beta-cell bursting can be maintained at up to 70% of beta-cell loss, which is consistent with laboratory and clinical findings of islets. Our results suggest that the bursting characteristics of a beta-cell cluster depend quantitatively on its architecture in a non-linear fashion. These findings are important to understand the islet bursting phenomenon and the regulation of insulin secretion, under both physiological and pathological conditions.