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

A primer on modelling pancreatic islets: from models of coupled β-cells to multicellular islet models

Pancreatic islets are mini-organs composed of hundreds or thousands of ɑ, β and δ-cells, which, respectively, secrete glucagon, insulin and somatostatin, key hormones for the regulation of blood glucose. In pancreatic islets, hormone secretion is tightly regulated by both internal and external mechanisms, including electrical communication and paracrine signaling between islet cells. Given its complexity, the experimental study of pancreatic islets has been complemented with computational modeling as a tool to gain a better understanding about how all the mechanisms involved at different levels of organization interact. In this review, we describe how multicellular models of pancreatic cells have evolved from the early models of electrically coupled β-cells to models in which experimentally derived architectures and both electrical and paracrine signals have been considered.

Pancreatic α and β cells are globally phase-locked

The Ca²⁺ modulated pulsatile glucagon and insulin secretions by pancreatic α and β cells play a crucial role in glucose homeostasis. However, how α and β cells coordinate to produce various Ca²⁺ oscillation patterns is still elusive. Using a microfluidic device and transgenic mice, we recorded Ca²⁺ signals from islet α and β cells, and observed heterogeneous Ca²⁺ oscillation patterns intrinsic to each islet. After a brief period of glucose stimulation, α and β cells’ oscillations were globally phase-locked. While the activation of α cells displayed a fixed time delay of ~20 s to that of β cells, β cells activated with a tunable period. Moreover, islet α cell number correlated with oscillation frequency. We built a mathematical model of islet Ca²⁺ oscillation incorporating paracrine interactions, which quantitatively agreed with the experimental data. Our study highlights the importance of cell-cell interaction in generating stable but tunable islet oscillation patterns.

The Ca²⁺ modulated pulsatile glucagon and insulin secretions by pancreatic α and β cells are critical in glucose homeostasis. Here the authors show that the Ca²⁺ oscillations of α and β cells are phase-locked, and that the oscillation pattern is tuned by paracrine interactions between α and β cells.

The physiological role of β-cell heterogeneity in pancreatic islet function

Endocrine cells within the pancreatic islets of Langerhans are heterogeneous in terms of transcriptional profile, protein expression and the regulation of hormone release. Even though this heterogeneity has long been appreciated, only within the past 5 years have detailed molecular analyses led to an improved understanding of its basis. Although we are beginning to recognize why some subpopulations of endocrine cells are phenotypically different to others, arguably the most important consideration is how this heterogeneity affects the regulation of hormone release to control the homeostasis of glucose and other energy-rich nutrients. The focus of this Review is the description of how endocrine cell heterogeneity (and principally that of insulin-secreting β-cells) affects the regulation of hormone secretion within the islets of Langerhans. This discussion includes an overview of the functional characteristics of the different islet cell subpopulations and describes how they can communicate to influence islet function under basal and glucose-stimulated conditions. We further discuss how changes to the specific islet cell subpopulations or their numbers might underlie islet dysfunction in type 2 diabetes mellitus. We conclude with a discussion of several key open questions regarding the physiological role of islet cell heterogeneity.

Small subpopulations of β-cells do not drive islet oscillatory [Ca2+] dynamics via gap junction communication

The islets of Langerhans exist as multicellular networks that regulate blood glucose levels. The majority of cells in the islet are excitable, insulin-producing β-cells that are electrically coupled via gap junction channels. β-cells are known to display heterogeneous functionality. However, due to gap junction coupling, β-cells show coordinated [Ca²⁺] oscillations when stimulated with glucose, and global quiescence when unstimulated. Small subpopulations of highly functional β-cells have been suggested to control [Ca²⁺] dynamics across the islet. When these populations were targeted by optogenetic silencing or photoablation, [Ca²⁺] dynamics across the islet were largely disrupted. In this study, we investigated the theoretical basis of these experiments and how small populations can disproportionality control islet [Ca²⁺] dynamics. Using a multicellular islet model, we generated normal, skewed or bimodal distributions of β-cell heterogeneity. We examined how islet [Ca²⁺] dynamics were disrupted when cells were targeted via hyperpolarization or populations were removed; to mimic optogenetic silencing or photoablation, respectively. Targeted cell populations were chosen based on characteristics linked to functional subpopulation, including metabolic rate of glucose oxidation or [Ca²⁺] oscillation frequency. Islets were susceptible to marked suppression of [Ca²⁺] when ~10% of cells with high metabolic activity were hyperpolarized; where hyperpolarizing cells with normal metabolic activity had little effect. However, when highly metabolic cells were removed from the model, [Ca²⁺] oscillations remained. Similarly, when ~10% of cells with either the highest frequency or earliest elevations in [Ca²⁺] were removed from the islet, the [Ca²⁺] oscillation frequency remained largely unchanged. Overall, these results indicate small populations of β-cells with either increased metabolic activity or increased frequency are unable to disproportionately control islet-wide [Ca²⁺] via gap junction coupling. Therefore, we need to reconsider the physiological basis for such small β-cell populations or the mechanism by which they may be acting to control normal islet function.

Many biological systems can be studied using network theory. How heterogeneous cell subpopulations come together to create complex multicellular behavior is of great value in understanding function and dysfunction in tissues. The pancreatic islet of Langerhans is a highly coupled structure that is important for maintaining blood glucose homeostasis. β-cell electrical activity is coordinated via gap junction communication. The function of the insulin-producing β-cell within the islet is disrupted in diabetes. As such, to understand the causes of islet dysfunction we need to understand how different cells within the islet contribute to its overall function via gap junction coupling. Using a computational model of β-cell electrophysiology, we investigated how small highly functional β-cell populations within the islet contribute to its function. We found that when small populations with greater functionality were introduced into the islet, they displayed signatures of this enhanced functionality. However, when these cells were removed, the islet, retained near-normal function. Thus, in a highly coupled system, such as an islet, the heterogeneity of cells allows small subpopulations to be dispensable, and thus their absence is unable to disrupt the larger cellular network. These findings can be applied to other electrical systems that have heterogeneous cell populations.

How Heterogeneity in Glucokinase and Gap-Junction Coupling Determines the Islet [Ca2+] Response

Understanding how cell subpopulations in a tissue impact overall system function is challenging. There is extensive heterogeneity among insulin-secreting β-cells within islets of Langerhans, including their insulin secretory response and gene expression profile, and this heterogeneity can be altered in diabetes. Several studies have identified variations in nutrient sensing between β-cells, including glucokinase (GK) levels, mitochondrial function, or expression of genes important for glucose metabolism. Subpopulations of β-cells with defined electrical properties can disproportionately influence islet-wide free-calcium activity ([Ca2+]) and insulin secretion via gap-junction electrical coupling. However, it is poorly understood how subpopulations of β-cells with altered glucose metabolism may impact islet function. To address this, we utilized a multicellular computational model of the islet in which a population of cells deficient in GK activity and glucose metabolism was imposed on the islet or in which β-cells were heterogeneous in glucose metabolism and GK kinetics were altered. This included simulating GK gene (GCK) mutations that cause monogenic diabetes. We combined these approaches with experimental models in which gck was genetically deleted in a population of cells or GK was pharmacologically inhibited. In each case, we modulated gap-junction electrical coupling. Both the simulated islet and the experimental system required 30-50% of the cells to have near-normal glucose metabolism, fewer than cells with normal KATP conductance. Below this number, the islet lacked any glucose-stimulated [Ca2+] elevations. In the absence of electrical coupling, the change in [Ca2+] was more gradual. As such, electrical coupling allows a large minority of cells with normal glucose metabolism to promote glucose-stimulated [Ca2+]. If insufficient numbers of cells are present, which we predict can be caused by a subset of GCK mutations that cause monogenic diabetes, electrical coupling exacerbates [Ca2+] suppression. This demonstrates precisely how metabolically heterogeneous β-cell populations interact to impact islet function.

Critical and Supercritical Spatiotemporal Calcium Dynamics in Beta Cells

A coordinated functioning of beta cells within pancreatic islets is mediated by oscillatory membrane depolarization and subsequent changes in cytoplasmic calcium concentration. While gap junctions allow for intraislet information exchange, beta cells within islets form complex syncytia that are intrinsically nonlinear and highly heterogeneous. To study spatiotemporal calcium dynamics within these syncytia, we make use of computational modeling and confocal high-speed functional multicellular imaging. We show that model predictions are in good agreement with experimental data, especially if a high degree of heterogeneity in the intercellular coupling term is assumed. In particular, during the first few minutes after stimulation, the probability distribution of calcium wave sizes is characterized by a power law, thus indicating critical behavior. After this period, the dynamics changes qualitatively such that the number of global intercellular calcium events increases to the point where the behavior becomes supercritical. To better mimic normal in vivo conditions, we compare the described behavior during supraphysiological non-oscillatory stimulation with the behavior during exposure to a slightly lower and oscillatory glucose challenge. In the case of this protocol, we observe only critical behavior in both experiment and model. Our results indicate that the loss of oscillatory changes, along with the rise in plasma glucose observed in diabetes, could be associated with a switch to supercritical calcium dynamics and loss of beta cell functionality.

The Pancreatic β-Cell: A Bioenergetic Perspective

The pancreatic β-cell secretes insulin in response to elevated plasma glucose. This review applies an external bioenergetic critique to the central processes of glucose-stimulated insulin secretion, including glycolytic and mitochondrial metabolism, the cytosolic adenine nucleotide pool, and its interaction with plasma membrane ion channels. The control mechanisms responsible for the unique responsiveness of the cell to glucose availability are discussed from bioenergetic and metabolic control standpoints. The concept of coupling factor facilitation of secretion is critiqued, and an attempt is made to unravel the bioenergetic basis of the oscillatory mechanisms controlling secretion. The need to consider the physiological constraints operating in the intact cell is emphasized throughout. The aim is to provide a coherent pathway through an extensive, complex, and sometimes bewildering literature, particularly for those unfamiliar with the field.

Gap junction proteins are key drivers of endocrine function

It has long been known that the main secretory cells of exocrine and endocrine glands are connected by gap junctions, made by a variety of connexin species that ensure their electrical and metabolic coupling. Experiments in culture systems and animal models have since provided increasing evidence that connexin signaling contributes to control the biosynthesis and release of secretory products, as well as to the life and death of secretory cells. More recently, genetic studies have further provided the first lines of evidence that connexins also control the function of human glands, which are central to the pathogenesis of major endocrine diseases. Here, we summarize the recent information gathered on connexin signaling in these systems, since the last reviews on the topic, with particular regard to the pancreatic beta cells which produce insulin, and the renal cells which produce renin. These cells are keys to the development of various forms of diabetes and hypertension, respectively, and combine to account for the exploding, worldwide prevalence of the metabolic syndrome. This article is part of a Special Issue entitled: Gap Junction Proteins edited by Jean Claude Herve.

Mathematical modeling of gap junction coupling and electrical activity in human β-cells

Coordinated insulin secretion is controlled by electrical coupling of pancreatic β-cells due to connexin-36 gap junctions. Gap junction coupling not only synchronizes the heterogeneous β-cell population, but can also modify the electrical behavior of the cells. These phenomena have been widely studied with mathematical models based on data from mouse β-cells. However, it is now known that human β-cell electrophysiology shows important differences to its rodent counterpart, and although human pancreatic islets express connexin-36 and show evidence of β-cell coupling, these aspects have been little investigated in human β-cells. Here we investigate theoretically, the gap junction coupling strength required for synchronizing electrical activity in a small cluster of cells simulated with a recent mathematical model of human β-cell electrophysiology. We find a lower limit for the coupling strength of approximately 20 pS (i.e., normalized to cell size, ∼2 pS pF(-1)) below which spiking electrical activity is asynchronous. To confront this theoretical lower bound with data, we use our model to estimate from an experimental patch clamp recording that the coupling strength is approximately 100-200 pS (10-20 pS pF(-1)), similar to previous estimates in mouse β-cells. We then investigate the role of gap junction coupling in synchronizing and modifying other forms of electrical activity in human β-cell clusters. We find that electrical coupling can prolong the period of rapid bursting electrical activity, and synchronize metabolically driven slow bursting, in particular when the metabolic oscillators are in phase. Our results show that realistic coupling conductances are sufficient to promote synchrony in small clusters of human β-cells as observed experimentally, and provide motivation for further detailed studies of electrical coupling in human pancreatic islets.

Intrinsic islet heterogeneity and gap junction coupling determine spatiotemporal Ca²⁺ wave dynamics

Insulin is released from the islets of Langerhans in discrete pulses that are linked to synchronized oscillations of intracellular free calcium ([Ca(2+)]i). Associated with each synchronized oscillation is a propagating calcium wave mediated by Connexin36 (Cx36) gap junctions. A computational islet model predicted that waves emerge due to heterogeneity in β-cell function throughout the islet. To test this, we applied defined patterns of glucose stimulation across the islet using a microfluidic device and measured how these perturbations affect calcium wave propagation. We further investigated how gap junction coupling regulates spatiotemporal [Ca(2+)]i dynamics in the face of heterogeneous glucose stimulation. Calcium waves were found to originate in regions of the islet having elevated excitability, and this heterogeneity is an intrinsic property of islet β-cells. The extent of [Ca(2+)]i elevation across the islet in the presence of heterogeneity is gap-junction dependent, which reveals a glucose dependence of gap junction coupling. To better describe these observations, we had to modify the computational islet model to consider the electrochemical gradient between neighboring β-cells. These results reveal how the spatiotemporal [Ca(2+)]i dynamics of the islet depend on β-cell heterogeneity and cell-cell coupling, and are important for understanding the regulation of coordinated insulin release across the islet.

Dimensionality and size scaling of coordinated Ca(2+) dynamics in MIN6 β-cell clusters

Pancreatic islets of Langerhans regulate blood glucose homeostasis by the secretion of the hormone insulin. Like many neuroendocrine cells, the coupling between insulin-secreting β-cells in the islet is critical for the dynamics of hormone secretion. We have examined how this coupling architecture regulates the electrical dynamics that underlie insulin secretion by utilizing a microwell-based aggregation method to generate clusters of a β-cell line with defined sizes and dimensions. We measured the dynamics of free-calcium activity ([Ca(2+)]i) and insulin secretion and compared these measurements with a percolating network model. We observed that the coupling dimension was critical for regulating [Ca(2+)]i dynamics and insulin secretion. Three-dimensional coupling led to size-invariant suppression of [Ca(2+)]i at low glucose and robust synchronized [Ca(2+)]i oscillations at elevated glucose, whereas two-dimensional coupling showed poor suppression and less robust synchronization, with significant size-dependence. The dimension- and size-scaling of [Ca(2+)]i at high and low glucose could be accurately described with the percolating network model, using similar network connectivity. As such this could explain the fundamentally different behavior and size-scaling observed under each coupling dimension. This study highlights the dependence of proper β-cell function on the coupling architecture that will be important for developing therapeutic treatments for diabetes such as islet transplantation techniques. Furthermore, this will be vital to gain a better understanding of the general features by which cellular interactions regulate coupled multicellular systems.

Cellular communication and heterogeneity in pancreatic islet insulin secretion dynamics

Coordinated pulses of electrical activity and insulin secretion are a hallmark of the islet of Langerhans. These coordinated behaviors are lost when β cells are dissociated, which also leads to increased insulin secretion at low glucose levels. Islets without gap junctions exhibit asynchronous electrical activity similar to dispersed cells, but their secretion at low glucose levels is still clamped off, putatively by a juxtacrine mechanism. Mice lacking β cell gap junctions have near-normal average insulin levels, but are glucose intolerant due to reduced first-phase and pulsatile insulin secretion, illustrating the importance of temporal dynamics. Here, we review the quantitative data on islet synchronization and the current mathematical models that have been developed to explain these behaviors and generate greater understanding of the underlying mechanisms.

The effect of unhealthy β-cells on insulin secretion in pancreatic islets

Background

Insulin secreted by pancreatic islet β-cells is the principal regulating hormone of glucose metabolism and plays a key role in controlling glucose level in blood. Impairment of the pancreatic islet function may cause glucose to accumulate in blood, and result in diabetes mellitus. Recent studies have shown that mitochondrial dysfunction has a strong negative effect on insulin secretion.

Methods

In order to study the cause of dysfunction of pancreatic islets, a multiple cell model containing healthy and unhealthy cells is proposed based on an existing single cell model. A parameter that represents the function of mitochondria is modified for unhealthy cells. A 3-D hexagonal lattice structure is used to model the spatial differences among β-cells in a pancreatic islet. The β-cells in the model are connected through direct electrical connections between neighboring β-cells.

Results

The simulation results show that the low ratio of total mitochondrial volume over cytoplasm volume per β-cell is a main reason that causes some mitochondria to lose their function. The results also show that the overall insulin secretion will be seriously disrupted when more than 15% of the β-cells in pancreatic islets become unhealthy.

Conclusion

Analysis of the model shows that the insulin secretion can be reinstated by increasing the glucokinase level. This new discovery sheds light on antidiabetic medication.

Phase transitions in pancreatic islet cellular networks and implications for type-1 diabetes

In many aspects the onset of a chronic disease resembles a phase transition in a complex dynamic system: Quantitative changes accumulate largely unnoticed until a critical threshold is reached, which causes abrupt qualitative changes of the system. In this study we examine a special case, the onset of type-1 diabetes (T1D), a disease that results from loss of the insulin-producing pancreatic islet β cells. Within each islet, the β cells are electrically coupled to each other via gap-junctional channels. This intercellular coupling enables the β cells to synchronize their insulin release, thereby generating the multiscale temporal rhythms in blood insulin that are critical to maintaining blood glucose homeostasis. Using percolation theory we show how normal islet function is intrinsically linked to network connectivity. In particular, the critical amount of β-cell death at which the islet cellular network loses site percolation is consistent with laboratory and clinical observations of the threshold loss of β cells that causes islet functional failure. In addition, numerical simulations confirm that the islet cellular network needs to be percolated for β cells to synchronize. Furthermore, the interplay between site percolation and bond strength predicts the existence of a transient phase of islet functional recovery after onset of T1D and introduction of treatment, potentially explaining the honeymoon phenomenon. Based on these results, we hypothesize that the onset of T1D may be the result of a phase transition of the islet β-cell network.

Coupling and noise induced spiking-bursting transition in a parabolic bursting model

The transition from tonic spiking to bursting is an important dynamic process that carry physiologically relevant information. In this work, coupling and noise induced spiking-bursting transition is investigated in a parabolic bursting model with specific discussion on their cooperation effects. Fast/slow analysis shows that weak coupling may help to induce the bursting by changing the geometric property of the fast subsystem so that the original unstable periodical solution are stabilized. It turned out that noise can play the similar stabilization role and induce bursting at appropriate moderate intensity. However, their cooperation may either strengthen or weaken the overall effect depending on the choice of noise level.

Connexins: key mediators of endocrine function

The appearance of multicellular organisms imposed the development of several mechanisms for cell-to-cell communication, whereby different types of cells coordinate their function. Some of these mechanisms depend on the intercellular diffusion of signal molecules in the extracellular spaces, whereas others require cell-to-cell contact. Among the latter mechanisms, those provided by the proteins of the connexin family are widespread in most tissues. Connexin signaling is achieved via direct exchanges of cytosolic molecules between adjacent cells at gap junctions, for cell-to-cell coupling, and possibly also involves the formation of membrane "hemi-channels," for the extracellular release of cytosolic signals, direct interactions between connexins and other cell proteins, and coordinated influence on the expression of multiple genes. Connexin signaling appears to be an obligatory attribute of all multicellular exocrine and endocrine glands. Specifically, the experimental evidence we review here points to a direct participation of the Cx36 isoform in the function of the insulin-producing β-cells of the endocrine pancreas, and of the Cx40 isoform in the function of the renin-producing juxtaglomerular epithelioid cells of the kidney cortex.

Glucose metabolism, islet architecture, and genetic homogeneity in imprinting of [Ca2+](i) and insulin rhythms in mouse islets

We reported previously that islets isolated from individual, outbred Swiss-Webster mice displayed oscillations in intracellular calcium ([Ca²⁺]_i) that varied little between islets of a single mouse but considerably between mice, a phenomenon we termed “islet imprinting.” We have now confirmed and extended these findings in several respects. First, imprinting occurs in both inbred (C57BL/6J) as well as outbred mouse strains (Swiss-Webster; CD1). Second, imprinting was observed in NAD(P)H oscillations, indicating a metabolic component. Further, short-term exposure to a glucose-free solution, which transiently silenced [Ca²⁺]_i oscillations, reset the oscillatory patterns to a higher frequency. This suggests a key role for glucose metabolism in maintaining imprinting, as transiently suppressing the oscillations with diazoxide, a K_ATP-channel opener that blocks [Ca²⁺]_i influx downstream of glucose metabolism, did not change the imprinted patterns. Third, imprinting was not as readily observed at the level of single beta cells, as the [Ca²⁺]_i oscillations of single cells isolated from imprinted islets exhibited highly variable, and typically slower [Ca²⁺]_i oscillations. Lastly, to test whether the imprinted [Ca²⁺]_i patterns were of functional significance, a novel microchip platform was used to monitor insulin release from multiple islets in real time. Insulin release patterns correlated closely with [Ca²⁺]_i oscillations and showed significant mouse-to-mouse differences, indicating imprinting. These results indicate that islet imprinting is a general feature of islets and is likely to be of physiological significance. While islet imprinting did not depend on the genetic background of the mice, glucose metabolism and intact islet architecture may be important for the imprinting phenomenon.

Glucose-dependent and -independent electrical activity in islets of Langerhans of Psammomys obesus, an animal model of nutritionally induced obesity and diabetes

Glucose-induced insulin secretion from pancreatic beta-cells involves metabolism-induced membrane depolarization and voltage-dependent Ca(2+) influx. The electrical events in beta-cell glucose sensing have been studied intensely using mouse islets of Langerhans, but data from other species, including models of type 2 diabetes mellitus (T2DM), are lacking. In this work, we made intracellular recordings of electrical activity from cells within islets of the gerbil Psammomys obesus (fat sand rat), a model of dietary-induced T2DM. Most islet cells from lean, non-diabetic sand rats displayed glucose-induced, K(ATP) channel-dependent, oscillatory electrical activity that was similar to the classic "bursting" pattern of mouse beta-cells. However, the oscillations were slower in sand rat islets, and the dose-response curve of electrical activity versus glucose concentration was left-shifted. Of the non-bursting cells, some produced action potentials continuously, while others displayed electrical activity that was largely independent of glucose. The latter activity consisted of continuous or intermittent action potential firing, and persisted for long periods in the absence of glucose. The glucose-insensitive activity was suppressed by diazoxide, indicating that the cells expressed K(ATP) channels. Sand rat islets produced intracellular Ca(2+) oscillations reminiscent of the oscillatory electrical pattern observed in most cells, albeit with a longer period. Finally, we found that the glucose dependence of insulin secretion from sand rat islets closely paralleled that of the bursting electrical activity. We conclude that while subpopulations of K(ATP)-expressing cells in sand rat islets display heterogeneous electrical responses to glucose, insulin secretion most closely follows the oscillatory activity. The ease of recording membrane potential from sand rat islets makes this a useful model for studies of beta-cell electrical signaling during the development of T2DM.

Contributions of mathematical modeling of beta cells to the understanding of beta-cell oscillations and insulin secretion

Mathematical modeling of pancreatic beta cells has contributed significantly to the understanding of the mechanisms involved in glucose-stimulated insulin secretion (GSIS). Early models of insulin secretion built in the 1970s were phenomenological with little biological foundation for the proposed mechanisms. In the 1980s, models focused on identifying the regulation of bursting electrical activity known to be important for insulin secretion. The main result was to reject proposed mechanisms as new data emerged, but important results of the role of cell-to-cell coupling were also established. New models have been proposed that provide possible explanations for the occurrence of various patterns of bursting and calcium oscillations. In addition, modeling has played an important role in comparing competing effects of calcium on both NADH and adenosine 3'-5'-cyclic monophosphate levels. Models including modern cell biological results of the regulation of insulin containing granules and cell heterogeneity have appeared, providing updated versions of the early models proposed in the 1970s. These models, when coupled to electrophysiological- and calcium-based ones, have the prospect to aid in understanding the overall picture of GSIS. In addition, they might be useful for estimating in vivo beta-cell functioning. Beta-cell modeling will likely move closer to clinical applications, where it can be expected to play an important role, as it has and will, in understanding the complex oscillatory phenomena observed in beta cells and islets.

Gap junction coupling and calcium waves in the pancreatic islet

The pancreatic islet is a highly coupled, multicellular system that exhibits complex spatiotemporal electrical activity in response to elevated glucose levels. The emergent properties of islets, which differ from those arising in isolated islet cells, are believed to arise in part by gap junctional coupling, but the mechanisms through which this coupling occurs are poorly understood. To uncover these mechanisms, we have used both high-speed imaging and theoretical modeling of the electrical activity in pancreatic islets under a reduction in the gap junction mediated electrical coupling. Utilizing islets from a gap junction protein connexin 36 knockout mouse model together with chemical inhibitors, we can modulate the electrical coupling in the islet in a precise manner and quantify this modulation by electrophysiology measurements. We find that after a reduction in electrical coupling, calcium waves are slowed as well as disrupted, and the number of cells showing synchronous calcium oscillations is reduced. This behavior can be reproduced by computational modeling of a heterogeneous population of beta-cells with heterogeneous levels of electrical coupling. The resulting quantitative agreement between the data and analytical models of islet connectivity, using only a single free parameter, reveals the mechanistic underpinnings of the multicellular behavior of the islet.

Mathematical modelling in neuroendocrinology

In neuroendocrinology, mathematical modelling is about formalising our understanding of the behaviour of the complex biological systems with which we deal. Formulating our explanations mathematically ensures their logical consistency, and makes them open to structured analysis; it is a stringent test of their intellectual coherence. In addition, however, modellers are seeking to extend our understanding in new ways, by seeking novel, simple explanations for complex behaviour. Here we discuss some styles of modelling as they have been applied to neuroendocrine systems, and discuss some of their strengths and limitations.

Islet-cell-to-cell communication as basis for normal insulin secretion

The emergence of pancreatic islets has necessitated the development of a signalling system for the intra- and inter-islet coordination of beta cells. With evolution, this system has evolved into a complex regulatory network of partially cross-talking pathways, whereby individual cells sense the state of activity of their neighbours and, accordingly, regulate their own level of functioning. A consistent feature of this network in vertebrates is the expression of connexin (Cx)-36-made cell-to-cell channels, which cluster at gap junction domains of the cell membrane, and which adjacent beta cells use to share cytoplasmic ions and small metabolites within individual islets. This chapter reviews what is known about Cx36, and the mechanism whereby this beta-cell connexin significantly regulates insulin secretion. It further outlines other less established functions of the protein and evaluates its potential relevance for the development of novel therapeutic approaches to diabetes.

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 β cells in each islet. The functional role of islet β 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 β-cell clusters, including the fraction of cells able to burst f_b, the synchronization index λ of the bursting β 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_β, number of inter-β cell couplings of each β cell n_c, and the coupling strength g_c. We found that at low values of n_β, 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_β∼100, n_c∼6 and g_c∼200 pS. In addition, normal β-cell clusters are robust against significant perturbation to their architecture, including the presence of non-β cells or dead β cells. In clusters with n_β>∼100, coordinated β-cell bursting can be maintained at up to 70% of β-cell loss, which is consistent with laboratory and clinical findings of islets. Our results suggest that the bursting characteristics of a β-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.