A biophysical model of electrical activity in human β-cells

Deconstructing the integrated oscillator model for pancreatic β-cells

Electrical bursting oscillations in the β-cells of pancreatic islets have been a focus of investigation for more than fifty years. This has been aided by mathematical models, which are descendants of the pioneering Chay-Keizer model. This article describes the key biophysical and mathematical elements of this model, and then describes the path forward from there to the Integrated Oscillator Model (IOM). It is both a history and a deconstruction of the IOM that describes the various elements that have been added to the model over time, and the motivation for adding them. Finally, the article is a celebration of the 40th anniversary of the publication of the Chay-Keizer model.

Multilevel synchronization of human β-cells networks

β-cells within the endocrine pancreas are fundamental for glucose, lipid and protein homeostasis. Gap junctions between cells constitute the primary coupling mechanism through which cells synchronize their electrical and metabolic activities. This evidence is still only partially investigated through models and numerical simulations. In this contribution, we explore the effect of combined electrical and metabolic coupling in β-cell clusters using a detailed biophysical model. We add heterogeneity and stochasticity to realistically reproduce β-cell dynamics and study networks mimicking arrangements of β-cells within human pancreatic islets. Model simulations are performed over different couplings and heterogeneities, analyzing emerging synchronization at the membrane potential, calcium, and metabolites levels. To describe network synchronization, we use the formalism of multiplex networks and investigate functional network properties and multiplex synchronization motifs over the structural, electrical, and metabolic layers. Our results show that metabolic coupling can support slow wave propagation in human islets, that combined electrical and metabolic synchronization is realized in small aggregates, and that metabolic long-range correlation is more pronounced with respect to the electrical one.

Machine learning provides insight into models of heterogeneous electrical activity in human beta-cells

Understanding how heterogeneous cellular responses emerge from cell-to-cell variations in expression and function of subcellular components is of general interest. Here, we focus on human insulin-secreting beta-cells, which are believed to constitute a population in which heterogeneity is of physiological importance. We exploit recent single-cell electrophysiological data that allow biologically realistic population modeling of human beta-cells that accounts for cellular heterogeneity and correlation between ion channel parameters. To investigate how ion channels influence the dynamics of our updated mathematical model of human pancreatic beta-cells, we explore several machine learning techniques to determine which model parameters are important for determining the qualitative patterns of electrical activity of the model cells. As expected, K^＋ channels promote absence of activity, but once a cell is active, they increase the likelihood of having action potential firing. HERG channels were of great importance for determining cell behavior in most of the investigated scenarios. Fast bursting is influenced by the time scales of ion channel activation and, interestingly, by the type of Ca^2＋ channels coupled to BK channels in BK-CaV complexes. Slow, metabolically driven oscillations are promoted mostly by K(ATP) channels. In summary, combining population modeling with machine learning analysis provides insight into the model and generates new hypotheses to be investigated both experimentally, via simulations and through mathematical analysis.

The effects of beta-cell mass and function, intercellular coupling, and islet synchrony on [Formula: see text] dynamics

Type 2 diabetes (T2D) is a challenging metabolic disorder characterized by a substantial loss of [Formula: see text]-cell mass and alteration of [Formula: see text]-cell function in the islets of Langerhans, disrupting insulin secretion and glucose homeostasis. The mechanisms for deficiency in [Formula: see text]-cell mass and function during the hyperglycemia development and T2D pathogenesis are complex. To study the relative contribution of [Formula: see text]-cell mass to [Formula: see text]-cell function in T2D, we make use of a comprehensive electrophysiological model of human [Formula: see text]-cell clusters. We find that defect in [Formula: see text]-cell mass causes a functional decline in single [Formula: see text]-cell, impairment in intra-islet synchrony, and changes in the form of oscillatory patterns of membrane potential and intracellular [Formula: see text] concentration, which can lead to changes in insulin secretion dynamics and in insulin levels. The model demonstrates a good correspondence between suppression of synchronizing electrical activity and published experimental measurements. We then compare the role of gap junction-mediated electrical coupling with both [Formula: see text]-cell synchronization and metabolic coupling in the behavior of [Formula: see text] concentration dynamics within human islets. Our results indicate that inter-[Formula: see text]-cellular electrical coupling depicts a more important factor in shaping the physiological regulation of islet function and in human T2D. We further predict that varying the whole-cell conductance of delayed rectifier [Formula: see text] channels modifies oscillatory activity patterns of [Formula: see text]-cell population lacking intercellular coupling, which significantly affect [Formula: see text] concentration and insulin secretion.

Mata A, Godínez-Fernández JR. Reconstructing human pancreatic islet architectures using computational optimization

We outline a general methodology based on computational optimization and experimental data to reconstruct human pancreatic islet architectures. By using the nuclei coordinates of islet cells obtained through DAPI staining, cell types identified by immunostaining, and cell size distributions estimated from capacitance measurements, reconstructed islets composed of non-overlapping spherical cells were obtained through an iterative optimization procedure. In all cases, the reconstructed architectures included >99% of the experimental identified cells, each of them having a radius within the experimentally reported ranges. Given the wide use of mathematical modeling for the study of pancreatic cells, and recently, of cell-cell interactions within the pancreatic islets, the methodology here proposed, also capable of identifying cell-to-cell contacts, is aimed to provide with a framework for modeling and analyzing experimentally-based pancreatic islet architectures.

Electrophysiological models of the human pancreatic δ-cell: From single channels to the firing of action potentials

Minimal mathematical models were developed to describe the electrophysiological properties of human δ-cells. Markov models of single channels were first developed based on the analysis of electrophysiological data. Monte Carlo simulations of voltage-clamp experiments were performed in an iteratively optimization procedure to estimate the number of channels required to reproduce the main characteristics of the macroscopic currents recorded experimentally. A membrane model of the firing of action potentials was then developed based on the kinetic schemes of single channels and the number of channels estimated. We showed that macroscopic currents of human δ-cells can be reproduced by minimal models of single channels when the appropriate number of channels is considered. In addition, our simulations suggest that human δ-cells are capable of generating action potentials through the interaction of the ionic currents involved. Finally, we determined the relative contribution of the currents underlying the firing of action potentials in human pancreatic δ-cells, which allowed us to propose a qualitative model of an action potential in terms of the underlying ionic currents.

From Local to Global Modeling for Characterizing Calcium Dynamics and Their Effects on Electrical Activity and Exocytosis in Excitable Cells

Electrical activity in neurons and other excitable cells is a result of complex interactions between the system of ion channels, involving both global coupling (e.g., via voltage or bulk cytosolic Ca²⁺ concentration) of the channels, and local coupling in ion channel complexes (e.g., via local Ca²⁺ concentration surrounding Ca²⁺ channels (CaVs), the so-called Ca²⁺ nanodomains). We recently devised a model of large-conductance BK_Ca potassium currents, and hence BK_Ca–CaV complexes controlled locally by CaVs via Ca²⁺ nanodomains. We showed how different CaV types and BK_Ca–CaV stoichiometries affect whole-cell electrical behavior. Ca²⁺ nanodomains are also important for triggering exocytosis of hormone-containing granules, and in this regard, we implemented a strategy to characterize the local interactions between granules and CaVs. In this study, we coupled electrical and exocytosis models respecting the local effects via Ca²⁺ nanodomains. By simulating scenarios with BK_Ca–CaV complexes with different stoichiometries in pituitary cells, we achieved two main electrophysiological responses (continuous spiking or bursting) and investigated their effects on the downstream exocytosis process. By varying the number and distance of CaVs coupled with the granules, we found that bursting promotes exocytosis with faster rates than spiking. However, by normalizing to Ca²⁺ influx, we found that bursting is only slightly more efficient than spiking when CaVs are far away from granules, whereas no difference in efficiency between bursting and spiking is observed with close granule-CaV coupling.

Biophysical modeling of β-cells networks: Realistic architectures and heterogeneity effects

The β-cells dynamics is the regulator of insulin secretion in the pancreas, and its investigation is a central aspect in designing effective treatment strategies for diabetes. Despite great efforts, much is still unknown about the complex organization of such endocrine cells and realistic mathematical modeling represents a useful tool to elucidate key aspects of glucose control in humans. In this contribution, we study the human β-cells collective behaviour, by modeling their electric and metabolic coupling in a cluster, of size and architecture similar to human islets of Langerhans. We focus on the effect of coupling on various dynamics regimes observed in the islets, that are spiking and bursting on multiple timescales. In particular, we test the effect of hubs, that are highly glucose-sensitive β-cells, on the overall network dynamics, observing different modulation depending on the timescale of the dynamics. By properly taking into account the role of cells heterogeneity, recently emerged, our model effectively describes the effect of hubs on the synchronization of the islet response and the correlation of β-cells activity.

Computational modeling of the effect of temperature variations on human pancreatic β-cell activity

The effect of temperature variations on the pancreatic β-cell activity and the role of different model compartments in temperature sensing have been investigated using a computational modeling approach. The results of our study show that temperature variations by several degrees can change the dynamical states of the β-cell system. In addition, temperature variations can alter the characteristic features of the membrane voltage, which correlates with insulin secretion. Simulation results show that the ion channels such as the L-type calcium, the hERG potassium, sodium channels and the glycolysis pathway are the possible sites for sensing temperature variation. Results indicate that for a small temperature change, even though the frequency and amplitude of electrical activity are altered, the area under the membrane potential curve remains almost unchanged, which implies the existence of a thermoregulatory mechanism for preserving the amount of insulin secretion. Furthermore, the computational analysis shows that the β-cell electrical activity exhibits a bursting pattern in physiological temperature (37 °C) while in vitro studies reported almost the spiking activity at lower temperatures. Since hormone-secreting systems work more efficient in bursting mode, we propose that the pancreatic β-cell works better in the physiological temperature compared with the reference temperature (33 °C).

Decoding the regulatory mechanism of glucose and insulin induced phosphatidylinositol 3,4,5-trisphosphate dynamics in β-cells

In MIN6 pancreatic β-cells, glucose and insulin act in a synergistic manner to regulate the dynamics of Phosphatidylinositol (3,4,5)-trisphosphate (PIP₃). However, the precise regulatory mechanism behind such an experimentally observed synergy is poorly understood. In this article, we propose a phenomenological mathematical model for studying the glucose and insulin driven PIP₃ activation dynamics under various stimulatory conditions to unfold the mechanism responsible for the observed synergy. The modeling study reveals that the experimentally observed oscillation in PIP₃ dynamics with disparate time scales for different external glucose doses is mainly orchestrated by the complex dynamic regulation of cytosolic Ca²⁺ in β-cells. The model accounts for the dose-dependent activation of PIP₃ as a function of externally added insulin, and further shows that even in the absence of Ca²⁺ signaling, externally added glucose can still maintain a basal level of endogenous insulin secretion via the fatty acid metabolism pathway. Importantly, the model analysis suggests that the glucose mediated ROS (reactive oxygen species) activation often contributes considerably to the synergistic activation of PIP₃ by glucose and insulin in a context dependent manner. Under the physiological conditions that keep β-cells in an insulin responsive state, the effect of glucose induced ROS signaling plays a moderate role in PIP₃ activation. As β-cells approach an insulin resistant state, the glucose induced ROS signaling significantly affects the PIP₃ dynamics. Our findings provide a plausible mechanistic insight into the experimentally observed synergy, and can lead to novel therapeutic strategies.

Investigation of the role of ion channels in human pancreatic β-cell hubs: A mathematical modeling study

In many cellular networks, the structure of the network follows a scale-free organization, where a limited number of cells are strongly coupled to other cells. These cells are called hub cells and their critical roles are well accepted. Despite their importance, there have been only a few studies investigating the characteristic features of these cells. In this paper, a computational approach is proposed to study the possible role of different ion channels in distinguishing between the hub and non-hub cells. The results show that the P/Q-type and T-type calcium channels may have an especial role in the β-cell hubs because the high-level expressions of these channels make a pancreatic β-cell more potent to force other coupled cells to follow it. In addition, in order to consider the variation of the coupling strength with voltage, a novel mathematical model is proposed for the gap junction coupling between the pancreatic β-cells. The proposed approach is validated based on the data from the literature.

Beta-cell hubs maintain Ca2+ oscillations in human and mouse islet simulations

Islet β-cells are responsible for secreting all circulating insulin in response to rising plasma glucose concentrations. These cells are a phenotypically diverse population that express great functional heterogeneity. In mice, certain β-cells (termed 'hubs') have been shown to be crucial for dictating the islet response to high glucose, with inhibition of these hub cells abolishing the coordinated Ca2+ oscillations necessary for driving insulin secretion. These β-cell hubs were found to be highly metabolic and susceptible to pro-inflammatory and glucolipotoxic insults. In this study, we explored the importance of hub cells in human by constructing mathematical models of Ca2+ activity in human islets. Our simulations revealed that hubs dictate the coordinated Ca2+ response in both mouse and human islets; silencing a small proportion of hubs abolished whole-islet Ca2+ activity. We also observed that if hubs are assumed to be preferentially gap junction coupled, then the simulations better adhere to the available experimental data. Our simulations of 16 size-matched mouse and human islet architectures revealed that there are species differences in the role of hubs; Ca2+ activity in human islets was more vulnerable to hub inhibition than mouse islets. These simulation results not only substantiate the existence of β-cell hubs, but also suggest that hubs may be favorably coupled in the electrical and metabolic network of the islet, and that targeted destruction of these cells would greatly impair human islet function.

The triggering pathway to insulin secretion: Functional similarities and differences between the human and the mouse β cells and their translational relevance

In β cells, stimulation by metabolic, hormonal, neuronal, and pharmacological factors is coupled to secretion of insulin through different intracellular signaling pathways. Our knowledge about the molecular machinery supporting these pathways and the patterns of signals it generates comes mostly from rodent models, especially the laboratory mouse. The increased availability of human islets for research during the last few decades has yielded new insights into the specifics in signaling pathways leading to insulin secretion in humans. In this review, we follow the most central triggering pathway to insulin secretion from its very beginning when glucose enters the β cell to the calcium oscillations it produces to trigger fusion of insulin containing granules with the plasma membrane. Along the way, we describe the crucial building blocks that contribute to the flow of information and focus on their functional role in mice and humans and on their translational implications.

Interaction between pancreatic β cell and electromagnetic fields: A systematic study toward finding the natural frequency spectrum of β cell system

Interaction between biological systems and environmental electric or magnetic fields has gained attention during the past few decades. Although there are a lot of studies that have been conducted for investigating such interaction, the reported results are considerably inconsistent. Besides the complexity of biological systems, the important reason for such inconsistent results may arise due to different excitation protocols that have been applied in different experiments. In order to investigate carefully the way that external electric or magnetic fields interact with a biological system, the parameters of excitation, such as intensity or frequency, should be selected purposefully due to the influence of these parameters on the system response. In this study, pancreatic β cell, the main player of blood glucose regulating system, is considered and the study is focused on finding the natural frequency spectrum of the system using modeling approach. Natural frequencies of a system are important characteristics of the system when external excitation is applied. The result of this study can help researchers to select proper frequency parameter for electrical excitation of β cell system. The results show that there are two distinct frequency ranges for natural frequency of β cell system, which consist of extremely low (or near zero) and 100-750 kHz frequency ranges. There are experimental works on β cell exposure to electromagnetic fields that support such finding.

Gap-junction coupling and ATP-sensitive potassium channels in human β-cell clusters: Effects on emergent dynamics

The importance of gap-junction coupling between β cells in pancreatic islets is well established in mouse. Such ultrastructural connections synchronize cellular activity, confine biological heterogeneity, and enhance insulin pulsatility. Dysfunction of coupling has been associated with diabetes and altered β-cell function. However, the role of gap junctions between human β cells is still largely unexplored. By using patch-clamp recordings of β cells from human donors, we previously estimated electrical properties of these channels by mathematical modeling of pairs of human β cells. In this work we revise our estimate by modeling triplet configurations and larger heterogeneous clusters. We find that a coupling conductance in the range 0.005-0.020 nS/pF can reproduce experiments in almost all the simulated arrangements. We finally explore the consequence of gap-junction coupling of this magnitude between β cells with mutant variants of the ATP-sensitive potassium channels involved in some metabolic disorders and diabetic conditions, translating studies performed on rodents to the human case. Our results are finally discussed from the perspective of therapeutic strategies. In summary, modeling of more realistic clusters with more than two β cells slightly lowers our previous estimate of gap-junction conductance and gives rise to patterns that more closely resemble experimental traces.

Concise Whole-Cell Modeling of BKCa-CaV Activity Controlled by Local Coupling and Stoichiometry

Large-conductance Ca²⁺-dependent K⁺ (BK_Ca) channels are important regulators of electrical activity. These channels colocalize and form ion channel complexes with voltage-dependent Ca²⁺ (CaV) channels. Recent stochastic simulations of the BK_Ca-CaV complex with 1:1 stoichiometry have given important insight into the local control of BK_Ca channels by fluctuating nanodomains of Ca²⁺. However, such Monte Carlo simulations are computationally expensive, and are therefore not suitable for large-scale simulations of cellular electrical activity. In this work we extend the stochastic model to more realistic BK_Ca-CaV complexes with 1:n stoichiometry, and analyze the single-complex model with Markov chain theory. From the description of a single BK_Ca-CaV complex, using arguments based on timescale analysis, we derive a concise model of whole-cell BK_Ca currents, which can readily be analyzed and inserted into models of cellular electrical activity. We illustrate the usefulness of our results by inserting our BK_Ca description into previously published whole-cell models, and perform simulations of electrical activity in various cell types, which show that BK_Ca-CaV stoichiometry can affect whole-cell behavior substantially. Our work provides a simple formulation for the whole-cell BK_Ca current that respects local interactions in BK_Ca-CaV complexes, and indicates how local-global coupling of ion channels may affect cell behavior.

Requirements for multi-level systems pharmacology models to reach end-usage: the case of type 2 diabetes

We are currently in the middle of a major shift in biomedical research: unprecedented and rapidly growing amounts of data may be obtained today, from in vitro, in vivo and clinical studies, at molecular, physiological and clinical levels. To make use of these large-scale, multi-level datasets, corresponding multi-level mathematical models are needed, i.e. models that simultaneously capture multiple layers of the biological, physiological and disease-level organization (also referred to as quantitative systems pharmacology-QSP-models). However, today's multi-level models are not yet embedded in end-usage applications, neither in drug research and development nor in the clinic. Given the expectations and claims made historically, this seemingly slow adoption may seem surprising. Therefore, we herein consider a specific example-type 2 diabetes-and critically review the current status and identify key remaining steps for these models to become mainstream in the future. This overview reveals how, today, we may use models to ask scientific questions concerning, e.g., the cellular origin of insulin resistance, and how this translates to the whole-body level and short-term meal responses. However, before these multi-level models can become truly useful, they need to be linked with the capabilities of other important existing models, in order to make them 'personalized' (e.g. specific to certain patient phenotypes) and capable of describing long-term disease progression. To be useful in drug development, it is also critical that the developed models and their underlying data and assumptions are easily accessible. For clinical end-usage, in addition, model links to decision-support systems combined with the engagement of other disciplines are needed to create user-friendly and cost-efficient software packages.

Fractal Behavior of the Pancreatic β-Cell Near the Percolation Threshold: Effect of the KATP Channel On the Electrical Response

The molecular system built with true chemical bonds or strong molecular interaction can be described using conceptual mathematical tools. Modeling of the natural generated ionic currents on the human pancreatic β-cell activity had been already studied using complicated analytical models. In our present contribution, we prove the same using our simple electrical model. The ionic currents are associated with different proteins membrane channels (K-Ca, K(v), K(ATP), Ca(v)-L) and Na/Ca Exchanger (NCX). The proteins are Ohmic conductors and are modeled by conductance randomly distributed. Switches are placed in series with conductances in order to highlight the channel activity. However, the KATP channel activity is stimulated by glucose, and the NCX's conductance change according to the intracellular calcium concentration. The percolation threshold of the system is calculated by the fractal nature of the infinite cluster using the Tarjan's depth-first-search algorithm. It is shown that the behavior of the internal concentration of Ca(2+) and the membrane potential are modulated by glucose. The results confirm that the inhibition of KATP channels depolarizes the membrane and increases the influx of [Ca(2+)]i through NCX and Ca(v)-L channel for high glucose concentrations.

Mathematical Modeling of Interacting Glucose-Sensing Mechanisms and Electrical Activity Underlying Glucagon-Like Peptide 1 Secretion

Intestinal L-cells sense glucose and other nutrients, and in response release glucagon-like peptide 1 (GLP-1), peptide YY and other hormones with anti-diabetic and weight-reducing effects. The stimulus-secretion pathway in L-cells is still poorly understood, although it is known that GLP-1 secreting cells use sodium-glucose co-transporters (SGLT) and ATP-sensitive K+-channels (K(ATP)-channels) to sense intestinal glucose levels. Electrical activity then transduces glucose sensing to Ca2+-stimulated exocytosis. This particular glucose-sensing arrangement with glucose triggering both a depolarizing SGLT current as well as leading to closure of the hyperpolarizing K(ATP) current is of more general interest for our understanding of glucose-sensing cells. To dissect the interactions of these two glucose-sensing mechanisms, we build a mathematical model of electrical activity underlying GLP-1 secretion. Two sets of model parameters are presented: one set represents primary mouse colonic L-cells; the other set is based on data from the GLP-1 secreting GLUTag cell line. The model is then used to obtain insight into the differences in glucose-sensing between primary L-cells and GLUTag cells. Our results illuminate how the two glucose-sensing mechanisms interact, and suggest that the depolarizing effect of SGLT currents is modulated by K(ATP)-channel activity. Based on our simulations, we propose that primary L-cells encode the glucose signal as changes in action potential amplitude, whereas GLUTag cells rely mainly on frequency modulation. The model should be useful for further basic, pharmacological and theoretical investigations of the cellular signals underlying endogenous GLP-1 and peptide YY release.

Modeling Ca(2+) currents and buffered diffusion of Ca(2+) in human β-cells during voltage clamp experiments

Macroscopic Ca(2+) currents of the human β-cells were characterized using the Hodgkin-Huxley formalism. Expressions describing the Ca(2+)-dependent inactivation process of the L-type Ca(2+) channels in terms of the concentration of Ca(2+) were obtained. By coupling the modeled Ca(2+) currents to a three-dimensional model of buffered diffusion of Ca(2+), we simulated the Ca(2+) transients formed in the immediate vicinity of the cell membrane during voltage clamp experiments performed in high buffering conditions. Our modeling approach allowed us to consider the distribution of the Ca(2+) sources over the cell membrane. The effect of exogenous (EGTA) and endogenous Ca(2+) buffers on the temporal course of the Ca(2+) transients was evaluated. We show that despite the high Ca(2+) buffering capacity, nanodomains are formed in the submembrane space, where a peak Ca(2+) concentration between ∼76 and 143 µM was estimated from our simulations. In addition, the contribution of each Ca(2+) current to the formation of the Ca(2+) nanodomains was also addressed. Here we provide a general framework to incorporate the spatial aspects to the models of the pancreatic β-cell, such as a more detailed and realistic description of Ca(2+) dynamics in response to electrical activity in physiological conditions can be provided by future models.

Role of topology in complex functional networks of beta cells

The activity of pancreatic β cells can be described by biological networks of coupled nonlinear oscillators that, via electrochemical synchronization, release insulin in response to augmented glucose levels. In this work, we analyze the emergent behavior of regular and percolated β-cells clusters through a stochastic mathematical model where "functional" networks arise. We show that the emergence and robustness of the synchronized dynamics depend both on intrinsic and extrinsic parameters. In particular, cellular noise level, glucose concentration, network spatial architecture, and cell-to-cell coupling strength are the key factors for the generation of a rhythmic and robust activity. Their role in the functional network topology associated with β-cells clusters is analyzed and discussed.

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.

Calcium and Metabolic Oscillations in Pancreatic Islets: Who's Driving the Bus?*

Pancreatic islets exhibit bursting oscillations in response to elevated blood glucose. These oscillations are accompanied by oscillations in the free cytosolic Ca2+ concentration (Cac ), which drives pulses of insulin secretion. Both islet Ca2+ and metabolism oscillate, but there is some debate about their interrelationship. Recent experimental data show that metabolic oscillations in some cases persist after the addition of diazoxide (Dz), which opens K(ATP) channels, hyperpolarizing β-cells and preventing Ca2+ entry and Ca2+ oscillations. Further, in some islets in which metabolic oscillations were eliminated with Dz, increasing the cytosolic Ca2+ concentration by the addition of KCl could restart the metabolic oscillations. Here we address why metabolic oscillations persist in some islets but not others, and why raising Cac restarts oscillations in some islets but not others. We answer these questions using the dual oscillator model (DOM) for pancreatic islets. The DOM can reproduce the experimental data and shows that the model supports two different mechanisms for slow metabolic oscillations, one that requires calcium oscillations and one that does not.

Pulsatile insulin secretion, impaired glucose tolerance and type 2 diabetes

Type 2 diabetes (T2DM) results when increases in beta cell function and/or mass cannot compensate for rising insulin resistance. Numerous studies have documented the longitudinal changes in metabolism that occur during the development of glucose intolerance and lead to T2DM. However, the role of changes in insulin secretion, both amount and temporal pattern, has been understudied. Most of the insulin secreted from pancreatic beta cells of the pancreas is released in a pulsatile pattern, which is disrupted in T2DM. Here we review the evidence that changes in beta cell pulsatility occur during the progression from glucose intolerance to T2DM in humans, and contribute significantly to the etiology of the disease. We review the evidence that insulin pulsatility improves the efficacy of secreted insulin on its targets, particularly hepatic glucose production, but also examine evidence that pulsatility alters or is altered by changes in peripheral glucose uptake. Finally, we summarize our current understanding of the biophysical mechanisms responsible for oscillatory insulin secretion. Understanding how insulin pulsatility contributes to normal glucose homeostasis and is altered in metabolic disease states may help improve the treatment of T2DM.

Reactive oxygen and nitrogen species disturb Ca(2+) oscillations in insulin-secreting MIN6 β-cells

Disturbances in pulsatile insulin secretion and Ca(2+) oscillations in pancreatic β-cells are early markers of diabetes, but the underlying mechanisms are still incompletely understood. Reactive oxygen/nitrogen species (ROS/RNS) are implicated in reduced β-cell function, and ROS/RNS target several Ca(2+) pumps and channels. Thus, we hypothesized that ROS/RNS could disturb Ca(2+) oscillations and downstream insulin pulsatility. We show that ROS/RNS production by photoactivation of aluminum phthalocyanine chloride (AlClPc) abolish or accelerate Ca(2+) oscillations in the MIN6 β-cell line, depending on the amount of ROS/RNS. Application of the sarcoplasmic/endoplasmic reticulum Ca(2+) ATPase (SERCA) inhibitor thapsigargin modifies the Ca(2+) response to high concentrations of ROS/RNS. Further, thapsigargin produces effects that resemble those elicited by moderate ROS/RNS production. These results indicate that ROS/RNS interfere with endoplasmic reticulum Ca(2+) handling. This idea is supported by theoretical studies using a mathematical model of Ca(2+) handling adapted to MIN6 cells. Our results suggest a putative link between ROS/RNS and disturbed pulsatile insulin secretion.

Calcium current inactivation rather than pool depletion explains reduced exocytotic rate with prolonged stimulation in insulin-secreting INS-1 832/13 cells

Impairment in beta-cell exocytosis is associated with reduced insulin secretion and diabetes. Here we aimed to investigate the dynamics of Ca2+-dependent insulin exocytosis with respect to pool depletion and Ca2+-current inactivation. We studied exocytosis, measured as increase in membrane capacitance (ΔCm), as a function of calcium entry (Q) in insulin secreting INS-1 832/13 cells using patch clamp and mixed-effects statistical analysis. The observed linear relationship between ΔCm and Q suggests that Ca2+-channel inactivation rather than granule pool restrictions is responsible for the decline in exocytosis observed at longer depolarizations. INS-1 832/13 cells possess an immediately releasable pool (IRP) of ∼10 granules and most exocytosis of granules occurs from a large pool. The latter is attenuated by the calcium-buffer EGTA, while IRP is unaffected. These findings suggest that most insulin release occurs away from Ca2+-channels, and that pool depletion plays a minor role in the decline of exocytosis upon prolonged stimulation.

Prediction model for glucose metabolism based on lipid metabolism

OBJECTIVES

We developed a robust, long-term clinical prediction model to predict conditions leading to early diabetes using laboratory values other than blood glucose and insulin levels. Our model protects against missing data and noise that occur during long-term analysis.

METHODS

RESULTS of a 75-g oral glucose tolerance test (OGTT) were divided into three groups: diabetes, impaired glucose tolerance (IGT), and normal (n = 114, 235, and 325, respectively). For glucose metabolic and lipid metabolic parameters, near 30-day mean values and 10-year integrated values were compared. The relation between high-density lipoprotein cholesterol (HDL-C) and variations in HbA1c was analyzed in 158 patients. We also constructed a state space model consisting of an observation model (HDL-C and HbA1c) and an internal model (disorders of lipid metabolism and glucose metabolism) and applied this model to 116 cases.

RESULTS

The root mean square error between the observed HbA1c and predicted HbA1c was 0.25.

CONCLUSIONS

In the observation model, HDL-C levels were useful for prediction of increases in HbA1c. Even with numerous missing values over time, as occurs in clinical practice, clinically valid predictions can be made using this state space model.

Multiscale modelling of insulin secretion during an intravenous glucose tolerance test

Dysfunctional insulin secretion from pancreatic β-cells plays a major role in the development of diabetes. The intravenous glucose tolerance test (IVGTT) is a widely used clinical test to assess β-cell function. The analysis of IVGTT data is conveniently performed using mathematical models, which need to be fairly simple to enable parameter identifiability (minimal models), but should at the same time have sound biological foundation at the cellular level. Using mathematical analysis and model reduction, we show here that our recent mathematical model of insulin secretory granule dynamics in β-cells provides mechanistic underpinning for our minimal model of pancreatic insulin secretion during an IVGTT.

Mathematical modeling of heterogeneous electrophysiological responses in human β-cells

Electrical activity plays a pivotal role in glucose-stimulated insulin secretion from pancreatic β-cells. Recent findings have shown that the electrophysiological characteristics of human β-cells differ from their rodent counterparts. We show that the electrophysiological responses in human β-cells to a range of ion channels antagonists are heterogeneous. In some cells, inhibition of small-conductance potassium currents has no effect on action potential firing, while it increases the firing frequency dramatically in other cells. Sodium channel block can sometimes reduce action potential amplitude, sometimes abolish electrical activity, and in some cells even change spiking electrical activity to rapid bursting. We show that, in contrast to L-type Ca2+-channels, P/Q-type Ca2+-currents are not necessary for action potential generation, and, surprisingly, a P/Q-type Ca2+-channel antagonist even accelerates action potential firing. By including SK-channels and Ca2+ dynamics in a previous mathematical model of electrical activity in human β-cells, we investigate the heterogeneous and nonintuitive electrophysiological responses to ion channel antagonists, and use our findings to obtain insight in previously published insulin secretion measurements. Using our model we also study paracrine signals, and simulate slow oscillations by adding a glycolytic oscillatory component to the electrophysiological model. The heterogenous electrophysiological responses in human β-cells must be taken into account for a deeper understanding of the mechanisms underlying insulin secretion in health and disease, and as shown here, the interdisciplinary combination of experiments and modeling increases our understanding of human β-cell physiology.

Mathematical models of electrical activity of the pancreatic β-cell: a physiological review

Mathematical modeling of the electrical activity of the pancreatic β-cell has been extremely important for understanding the cellular mechanisms involved in glucose-stimulated insulin secretion. Several models have been proposed over the last 30 y, growing in complexity as experimental evidence of the cellular mechanisms involved has become available. Almost all the models have been developed based on experimental data from rodents. However, given the many important differences between species, models of human β-cells have recently been developed. This review summarizes how modeling of β-cells has evolved, highlighting the proposed physiological mechanisms underlying β-cell electrical activity.

Control of voltage-gated potassium channel Kv2.2 expression by pyruvate-isocitrate cycling regulates glucose-stimulated insulin secretion

Recent studies have shown that the pyruvate-isocitrate cycling pathway, involving the mitochondrial citrate/isocitrate carrier and the cytosolic NADP-dependent isocitrate dehydrogenase (ICDc), is involved in control of glucose-stimulated insulin secretion (GSIS). Here we demonstrate that pyruvate-isocitrate cycling regulates expression of the voltage-gated potassium channel family member Kv2.2 in islet β-cells. siRNA-mediated suppression of ICDc, citrate/isocitrate carrier, or Kv2.2 expression impaired GSIS, and the effect of ICDc knockdown was rescued by re-expression of Kv2.2. Moreover, chronic exposure of β-cells to elevated fatty acids, which impairs GSIS, resulted in decreased expression of Kv2.2. Surprisingly, knockdown of ICDc or Kv2.2 increased rather than decreased outward K(+) current in the 832/13 β-cell line. Immunoprecipitation studies demonstrated interaction of Kv2.1 and Kv2.2, and co-overexpression of the two channels reduced outward K(+) current compared with overexpression of Kv2.1 alone. Also, siRNA-mediated knockdown of ICDc enhanced the suppressive effect of the Kv2.1-selective inhibitor stromatoxin1 on K(+) currents. Our data support a model in which a key function of the pyruvate-isocitrate cycle is to maintain levels of Kv2.2 expression sufficient to allow it to serve as a negative regulator of Kv channel activity.

Ion channels and regulation of insulin secretion in human β-cells: a computational systems analysis

In mammals an increase in glucose leads to block of ATP dependent potassium channels in pancreatic β cells leading to membrane depolarization. This leads to the repetitive firing of action potentials that increases calcium influx and triggers insulin granule exocytosis. Several important differences between species in this process suggest that a dedicated human-oriented approach is advantageous as extrapolating from rodent data may be misleading in several respects. We examined depolarization-induced spike activity in pancreatic human islet-attached β-cells employing whole-cell patch-clamp methods. We also reviewed the literature concerning regulation of insulin secretion by channel activity and constructed a data-based computer model of human β cell function. The model couples the Hodgkin-Huxley-type ionic equations to the equations describing intracellular Ca²⁺ homeostasis and insulin release. On the basis of this model we employed computational simulations to better understand the behavior of action potentials, calcium handling and insulin secretion in human β cells under a wide range of experimental conditions. This computational system approach provides a framework to analyze the mechanisms of human β cell insulin secretion.

The voltage-dependent potassium channel subunit Kv2.1 regulates insulin secretion from rodent and human islets independently of its electrical function

AIMS/HYPOTHESIS

It is thought that the voltage-dependent potassium channel subunit Kv2.1 (Kv2.1) regulates insulin secretion by controlling beta cell electrical excitability. However, this role of Kv2.1 in human insulin secretion has been questioned. Interestingly, Kv2.1 can also regulate exocytosis through direct interaction of its C-terminus with the soluble NSF attachment receptor (SNARE) protein, syntaxin 1A. We hypothesised that this interaction mediates insulin secretion independently of Kv2.1 electrical function.

METHODS

Wild-type Kv2.1 or mutants lacking electrical function and syntaxin 1A binding were studied in rodent and human beta cells, and in INS-1 cells. Small intracellular fragments of the channel were used to disrupt native Kv2.1-syntaxin 1A complexes. Single-cell exocytosis and ion channel currents were monitored by patch-clamp electrophysiology. Interaction between Kv2.1, syntaxin 1A and other SNARE proteins was probed by immunoprecipitation. Whole-islet Ca(2+)-responses were monitored by ratiometric Fura red fluorescence and insulin secretion was measured.

RESULTS

Upregulation of Kv2.1 directly augmented beta cell exocytosis. This happened independently of channel electrical function, but was dependent on the Kv2.1 C-terminal syntaxin 1A-binding domain. Intracellular fragments of the Kv2.1 C-terminus disrupted native Kv2.1-syntaxin 1A interaction and impaired glucose-stimulated insulin secretion. This was not due to altered ion channel activity or impaired Ca(2+)-responses to glucose, but to reduced SNARE complex formation and Ca(2+)-dependent exocytosis.

CONCLUSIONS/INTERPRETATION

Direct interaction between syntaxin 1A and the Kv2.1 C-terminus is required for efficient insulin exocytosis and glucose-stimulated insulin secretion. This demonstrates that native Kv2.1-syntaxin 1A interaction plays a key role in human insulin secretion, which is separate from the channel's electrical function.

Membranes with the same ion channel populations but different excitabilities

Electrical signaling allows communication within and between different tissues and is necessary for the survival of multicellular organisms. The ionic transport that underlies transmembrane currents in cells is mediated by transporters and channels. Fast ionic transport through channels is typically modeled with a conductance-based formulation that describes current in terms of electrical drift without diffusion. In contrast, currents written in terms of drift and diffusion are not as widely used in the literature in spite of being more realistic and capable of displaying experimentally observable phenomena that conductance-based models cannot reproduce (e.g. rectification). The two formulations are mathematically related: conductance-based currents are linear approximations of drift-diffusion currents. However, conductance-based models of membrane potential are not first-order approximations of drift-diffusion models. Bifurcation analysis and numerical simulations show that the two approaches predict qualitatively and quantitatively different behaviors in the dynamics of membrane potential. For instance, two neuronal membrane models with identical populations of ion channels, one written with conductance-based currents, the other with drift-diffusion currents, undergo transitions into and out of repetitive oscillations through different mechanisms and for different levels of stimulation. These differences in excitability are observed in response to excitatory synaptic input, and across different levels of ion channel expression. In general, the electrophysiological profiles of membranes modeled with drift-diffusion and conductance-based models having identical ion channel populations are different, potentially causing the input-output and computational properties of networks constructed with these models to be different as well. The drift-diffusion formulation is thus proposed as a theoretical improvement over conductance-based models that may lead to more accurate predictions and interpretations of experimental data at the single cell and network levels.

Block of Kv1.7 potassium currents increases glucose-stimulated insulin secretion

Glucose-stimulated insulin secretion (GSIS) relies on repetitive, electrical spiking activity of the beta cell membrane. Cyclic activation of voltage-gated potassium channels (K_v) generates an outward, ‘delayed rectifier’ potassium current, which drives the repolarizing phase of each spike and modulates insulin release. Although several K_v channels are expressed in pancreatic islets, their individual contributions to GSIS remain incompletely understood. We take advantage of a naturally occurring cone-snail peptide toxin, Conkunitzin-S1 (Conk-S1), which selectively blocks K_v1.7 channels to provide an intrinsically limited, finely graded control of total beta cell delayed rectifier current and hence of GSIS. Conk-S1 increases GSIS in isolated rat islets, likely by reducing K_v1.7-mediated delayed rectifier currents in beta cells, which yields increases in action potential firing and cytoplasmic free calcium. In rats, Conk-S1 increases glucose-dependent insulin secretion without decreasing basal glucose. Thus, we conclude that K_v1.7 contributes to the membrane-repolarizing current of beta cells during GSIS and that block of this specific component of beta cell K_v current offers a potential strategy for enhancing GSIS with minimal risk of hypoglycaemia during metabolic disorders such as Type 2 diabetes.

Mathematical models for insulin secretion in pancreatic β-cells

Insulin secretion is one of the most characteristic features of β-cell physiology. As it plays a central role in glucose regulation, a number of experimental and theoretical studies have been performed since the discovery of the pancreatic β-cell. This review article aims to give an overview of the mathematical approaches to insulin secretion. Beginning with the bursting electrical activity in pancreatic β-cells, we describe effects of the gap-junction coupling between β-cells on the dynamics of insulin secretion. Then, implications of paracrine interactions among such islet cells as α-, β-, and δ-cells are discussed. Finally, we present mathematical models which incorporate effects of glycolysis and mitochondrial glucose metabolism on the control of insulin secretion.

Mathematical modeling demonstrates how multiple slow processes can provide adjustable control of islet bursting

Pancreatic islets exhibit bursting oscillations that give rise to oscillatory Ca (2+) entry and insulin secretion from β-cells. These oscillations are driven by a slowly activating K (+) current, Kslow, which is composed of two components: an ATP-sensitive K (+) current and a Ca (2+) -activated K (+) current through SK4 channels. Using a mathematical model of pancreatic β-cells, we analyze how the factors that comprise Kslow can contribute to bursting. We employ the dominance factor technique developed recently to do this and demonstrate that the contributions the slow processes make to bursting are non-obvious and often counterintuitive, and that their contributions vary with parameter values and are thus adjustable.

Mathematical modeling and statistical analysis of calcium-regulated insulin granule exocytosis in β-cells from mice and humans

Insulin is released from pancreatic β-cells as a result of Ca²⁺-evoked exocytosis of dense-core granules. Secretion is biphasic, which has been suggested to correspond to the release of different granule pools. Here we review and carefully reanalyze previously published patch-clamp data on depolarization-evoked Ca²⁺-currents and corresponding capacitance measurements. Using a statistical mixed-effects model, we show that the data indicate that pool depletion is negligible in response to short depolarizations in mouse β-cells. We then review mathematical models of granule dynamics and exocytosis in rodent β-cells and present a mathematical description of Ca²⁺-evoked exocytosis in human β-cells, which show clear differences to their rodent counterparts. The model suggests that L- and P/Q-type Ca²⁺-channels are involved to a similar degree in exocytosis during electrical activity in human β-cells.