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

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.

Amplitude-modulated spiking as a novel route to bursting: Coupling-induced mixed-mode oscillations by symmetry breaking

Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are increasingly well understood and are often caused by folded singularities, canard orbits, or singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for diffusively coupled FitzHugh-Nagumo oscillators with negative coupling constant, and we show that it is caused by a singular Hopf bifurcation related to a folded saddle-node (FSN) singularity. Inspired by earlier work on models of pancreatic beta-cells [Sherman, Bull. Math. Biol. 56, 811 (1994)], we then identify a new type of bursting dynamics due to diffusive coupling of cells firing action potentials when isolated. In the presence of coupling, small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. Confirming the hypothesis from the earlier work that this behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem, we find that it is caused by symmetry breaking. Moreover, we show that it is organized by a FSN in the averaged system, which causes a singular Hopf bifurcation. Such behavior is related to the recently studied dynamics caused by the so-called torus canards.

Modelling of energy-driven switch for glucagon and insulin secretion

We present a mathematical model of the energy-driven metabolic switch for glucagon and insulin secretion from pancreatic alpha and beta cells, respectively. The energy status related to hormone secretion is studied for various glucose concentrations. Additionally, the physiological response is studied with regards to the presence of other metabolites, particularly the free-fatty acids. At low glucose, the ATP production in alpha cells is high due to free-fatty acids oxidation in mitochondria, which enables glucagon secretion. When the glucose concentration is elevated above the threshold value, the glucagon secretion is switched off due to the contribution of glycolytic ATP production, representing an "anaerobic switch". On the other hand, during hypoglycemia, the ATP production in beta cells is low, reflecting a "waiting state" for glucose as the main metabolite. When glucose is elevated above the threshold value, the oxidative fate of glucose in mitochondria is the main source of energy required for effective insulin secretion, i.e. the "aerobic switch". Our results show the importance of well-regulated and fine-tuned energetic processes in pancreatic alpha and beta cells required for efficient hormone secretion and hence effective blood glucose regulation. These energetic processes have to be appropriately switched on and off based on the sensing of different metabolites by alpha and beta cells. Our computational results indicate that disturbances in cell energetics (e.g. mitochondrial dysfunction), and dysfunctional metabolite sensing and distribution throughout the cell might be related to pathologies such as metabolic syndrome and diabetes.

Grand canonical diffusion-influenced reactions: A stochastic theory with applications to multiscale reaction-diffusion simulations

Smoluchowski-type models for diffusion-influenced reactions (A + B → C) can be formulated within two frameworks: the probabilistic-based approach for a pair A, B of reacting particles and the concentration-based approach for systems in contact with a bath that generates a concentration gradient of B particles that interact with A. Although these two approaches are mathematically similar, it is not straightforward to establish a precise mathematical relationship between them. Determining this relationship is essential to derive particle-based numerical methods that are quantitatively consistent with bulk concentration dynamics. In this work, we determine the relationship between the two approaches by introducing the grand canonical Smoluchowski master equation (GC-SME), which consists of a continuous-time Markov chain that models an arbitrary number of B particles, each one of them following Smoluchowski's probabilistic dynamics. We show that the GC-SME recovers the concentration-based approach by taking either the hydrodynamic or the large copy number limit. In addition, we show that the GC-SME provides a clear statistical mechanical interpretation of the concentration-based approach and yields an emergent chemical potential for nonequilibrium spatially inhomogeneous reaction processes. We further exploit the GC-SME robust framework to accurately derive multiscale/hybrid numerical methods that couple particle-based reaction-diffusion simulations with bulk concentration descriptions, as described in detail through two computational implementations.

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.

COEXISTENCE OF THREE OSCILLATORY MODES OF INSULIN SECRETION: MATHEMATICAL MODELING AND RELEVANCE TO GLUCOSE REGULATION

Insulin secretion in pancreatic [Formula: see text]-cells exhibits three oscillatory modes with distinct period ranges, called fast, slow, and ultradian modes. To unveil the mechanism underlying such oscillatory behaviors and their roles in blood glucose regulation, we propose a combined model for the glucose–insulin regulation system, incorporating both the cell-level insulin secretion mechanism and inter-organ interactions in the blood glucose regulation. Special emphasis is placed on the identification of the mechanism of the slow oscillation and its role associated with the whole-body glucose regulation. Via extensive numerical simulations, we obtain macroscopic behaviors of the three types of insulin/glucose oscillations in the whole-body as well as microscopic behaviors of the membrane potential and the calcium concentration in the [Formula: see text]-cell. Finally, optimal regulatory strategies for the blood glucose level are discussed on the basis of the quantitative information obtained from the mathematical modeling and numerical simulations.

Ca2+ channel clustering with insulin-containing granules is disturbed in type 2 diabetes

Loss of first-phase insulin secretion is an early sign of developing type 2 diabetes (T2D). Ca²⁺ entry through voltage-gated L-type Ca²⁺ channels triggers exocytosis of insulin-containing granules in pancreatic β cells and is required for the postprandial spike in insulin secretion. Using high-resolution microscopy, we have identified a subset of docked insulin granules in human β cells and rat-derived clonal insulin 1 (INS1) cells for which localized Ca²⁺ influx triggers exocytosis with high probability and minimal latency. This immediately releasable pool (IRP) of granules, identified both structurally and functionally, was absent in β cells from human T2D donors and in INS1 cells cultured in fatty acids that mimic the diabetic state. Upon arrival at the plasma membrane, IRP granules slowly associated with 15 to 20 L-type channels. We determined that recruitment depended on a direct interaction with the synaptic protein Munc13, because expression of the II–III loop of the channel, the C2 domain of Munc13-1, or of Munc13-1 with a mutated C2 domain all disrupted L-type channel clustering at granules and ablated fast exocytosis. Thus, rapid insulin secretion requires Munc13-mediated recruitment of L-type Ca²⁺ channels in close proximity to insulin granules. Loss of this organization underlies disturbed insulin secretion kinetics in T2D.

Recent advances in mathematical modeling and statistical analysis of exocytosis in endocrine cells

Most endocrine cells secrete hormones as a result of Ca²⁺-regulated exocytosis, i.e., fusion of the membranes of hormone-containing secretory granules with the cell membrane, which allows the hormone molecules to escape to the extracellular space. As in neurons, electrical activity and cell depolarization open voltage-sensitive Ca²⁺ channels, and the resulting Ca²⁺ influx elevate the intracellular Ca²⁺ concentration, which in turn causes exocytosis. Whereas the main molecular components involved in exocytosis are increasingly well understood, quantitative understanding of the dynamical aspects of exocytosis is still lacking. Due to the nontrivial spatiotemporal Ca²⁺ dynamics, which depends on the particular pattern of electrical activity as well as Ca²⁺ channel kinetics, exocytosis is dependent on the spatial arrangement of Ca²⁺ channels and secretory granules. For example, the creation of local Ca²⁺ microdomains, where the Ca²⁺ concentration reaches tens of µM, are believed to be important for triggering exocytosis. Spatiotemporal simulations of buffered Ca²⁺ diffusion have provided important insight into the interplay between electrical activity, Ca²⁺ channel kinetics, and the location of granules and Ca²⁺ channels. By confronting simulations with statistical time-to-event (or survival) regression analysis of single granule exocytosis monitored with TIRF microscopy, a direct connection between location and rate of exocytosis can be obtained at the local, single-granule level. To get insight into whole-cell secretion, simplifications of the full spatiotemporal dynamics have shown to be highly helpful. Here, we provide an overview of recent approaches and results for quantitative analysis of Ca²⁺ regulated exocytosis of hormone-containing granules.

Pancreatic Beta Cell G-Protein Coupled Receptors and Second Messenger Interactions: A Systems Biology Computational Analysis

Insulin secretory in pancreatic beta-cells responses to nutrient stimuli and hormonal modulators include multiple messengers and signaling pathways with complex interdependencies. Here we present a computational model that incorporates recent data on glucose metabolism, plasma membrane potential, G-protein-coupled-receptors (GPCR), cytoplasmic and endoplasmic reticulum calcium dynamics, cAMP and phospholipase C pathways that regulate interactions between second messengers in pancreatic beta-cells. The values of key model parameters were inferred from published experimental data. The model gives a reasonable fit to important aspects of experimentally measured metabolic and second messenger concentrations and provides a framework for analyzing the role of metabolic, hormones and neurotransmitters changes on insulin secretion. Our analysis of the dynamic data provides support for the hypothesis that activation of Ca²⁺-dependent adenylyl cyclases play a critical role in modulating the effects of glucagon-like peptide 1 (GLP-1), glucose-dependent insulinotropic polypeptide (GIP) and catecholamines. The regulatory properties of adenylyl cyclase isoforms determine fluctuations in cytoplasmic cAMP concentration and reveal a synergistic action of glucose, GLP-1 and GIP on insulin secretion. On the other hand, the regulatory properties of phospholipase C isoforms determine the interaction of glucose, acetylcholine and free fatty acids (FFA) (that act through the FFA receptors) on insulin secretion. We found that a combination of GPCR agonists activating different messenger pathways can stimulate insulin secretion more effectively than a combination of GPCR agonists for a single pathway. This analysis also suggests that the activators of GLP-1, GIP and FFA receptors may have a relatively low risk of hypoglycemia in fasting conditions whereas an activator of muscarinic receptors can increase this risk. This computational analysis demonstrates that study of second messenger pathway interactions will improve understanding of critical regulatory sites, how different GPCRs interact and pharmacological targets for modulating insulin secretion in type 2 diabetes.

Is bursting more effective than spiking in evoking pituitary hormone secretion? A spatiotemporal simulation study of calcium and granule dynamics

Endocrine cells of the pituitary gland secrete a number of hormones, and the amount of hormone released by a cell is controlled in large part by the cell's electrical activity and subsequent Ca(2+) influx. Typical electrical behaviors of pituitary cells include continuous spiking and so-called pseudo-plateau bursting. It has been shown that the amplitude of Ca(2+) fluctuations is greater in bursting cells, leading to the hypothesis that bursting cells release more hormone than spiking cells. In this work, we apply computer simulations to test this hypothesis. We use experimental recordings of electrical activity as input to mathematical models of Ca(2+) channel activity, buffered Ca(2+) diffusion, and Ca(2+)-driven exocytosis. To compare the efficacy of spiking and bursting on the same cell, we pharmacologically block the large-conductance potassium (BK) current from a bursting cell or add a BK current to a spiking cell via dynamic clamp. We find that bursting is generally at least as effective as spiking at evoking hormone release and is often considerably more effective, even when normalizing to Ca(2+) influx. Our hybrid experimental/modeling approach confirms that adding a BK-type K(+) current, which is typically associated with decreased cell activity and reduced secretion, can actually produce an increase in hormone secretion, as suggested earlier.

Mathematical modelling of local calcium and regulated exocytosis during inhibition and stimulation of glucagon secretion from pancreatic alpha-cells

Glucagon secretion from pancreatic alpha-cells is dysregulated in diabetes. Despite decades of investigations of the control of glucagon release by glucose and hormones, the underlying mechanisms are still debated. Recently, mathematical models have been applied to investigate the modification of electrical activity in alpha-cells as a result of glucose application. However, recent studies have shown that paracrine effects such as inhibition of glucagon secretion by glucagon-like peptide 1 (GLP-1) or stimulation of release by adrenaline involve cAMP-mediated effects downstream of electrical activity. In particular, depending of the intracellular cAMP concentration, specific types of Ca(2+) channels are inhibited or activated, which interacts with mobilization of secretory granules. To investigate these aspects of alpha-cell function theoretically, we carefully developed a mathematical model of Ca(2+) levels near open or closed Ca(2+) channels of various types, which was linked to a description of Ca(2+) below the plasma membrane, in the bulk cytosol and in the endoplasmic reticulum. We investigated how the various subcellular Ca(2+) compartments contribute to control of glucagon-exocytosis in response to glucose, GLP-1 or adrenaline. Our studies refine previous modelling studies of alpha-cell function, and provide deeper insight into the control of glucagon 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.

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.

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.

A computational systems analysis of factors regulating α cell glucagon secretion

Glucagon, a peptide hormone secreted from the α-cells of the pancreatic islets, is critical for blood glucose homeostasis. We reviewed the literature and employed a computational systems analysis of intracellular metabolic and electrical regulation of glucagon secretion to better understand these processes. The mathematical model of α-cell metabolic parameters is based on our previous model for pancreatic β-cells. We also formulated an ionic model for action potentials that incorporates Ca ( 2+) , K (+) , Na (+) and Cl (-) currents. Metabolic and ionic models are coupled to the equations describing Ca ( 2+) homeostasis and glucagon secretion that depends on activation of specific voltage-gated Ca ( 2+) channels. Paracrine and endocrine regulations were analyzed with an emphasis on their effects on a hyperpolarization of membrane potential. This general model simulates and gives insight into the mechanisms of regulation of glucagon secretion under a wide range of experimental conditions. We also reviewed and analyzed dysfunctional mechanisms in α-cells to determine key pharmacological targets for modulating glucagon secretion in type 1 and 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.