Ca2+ Effects on ATP Production and Consumption Have Regulatory Roles on Oscillatory Islet Activity

Kinetic modelling of glycolytic oscillations

Glycolytic oscillations have been studied for well over 60 years, but aspects of their function, and mechanisms of regulation and synchronisation remain unclear. Glycolysis is amenable to mechanistic mathematical modelling, as its components have been well characterised, and the system can be studied at many organisational levels: in vitro reconstituted enzymes, cell free extracts, individual cells, and cell populations. In recent years, the emergence of individual cell analysis has opened new ways of studying this intriguing system.

TALK-1-mediated alterations of β-cell mitochondrial function and insulin secretion impair glucose homeostasis on a diabetogenic diet

Mitochondrial Ca2+ ([Ca2+]m) homeostasis is critical for β-cell function and becomes disrupted during the pathogenesis of diabetes. [Ca2+]m uptake is dependent on elevations in cytoplasmic Ca2+ ([Ca2+]c) and endoplasmic reticulum Ca2+ ([Ca2+]ER) release, both of which are regulated by the two-pore domain K+ channel TALK-1. Here, utilizing a novel β-cell TALK-1-knockout (β-TALK-1-KO) mouse model, we found that TALK-1 limited β-cell [Ca2+]m accumulation and ATP production. However, following exposure to a high-fat diet (HFD), ATP-linked respiration, glucose-stimulated oxygen consumption rate, and glucose-stimulated insulin secretion (GSIS) were increased in control but not TALK1-KO mice. Although β-TALK-1-KO animals showed similar GSIS before and after HFD treatment, these mice were protected from HFD-induced glucose intolerance. Collectively, these data identify that TALK-1 channel control of β-cell function reduces [Ca2+]m and suggest that metabolic remodeling in diabetes drives dysglycemia.

Periodic insulin stimulation of Akt: Dynamic steady states and robustness

The periodic secretion of insulin is a salient feature of the blood glucose control system in vivo. Insulin levels in the blood exhibit oscillations on multiple time scales - rapid, ultradian, and circadian - and the improved metabolic regulation resulting from pulsatile insulin release has been well established. Although numerous mathematical models investigating the causal mechanisms of insulin oscillations have appeared in the literature, to date there has been comparatively little attention given to the influence of periodic insulin stimulation on downstream components of the insulin signalling pathway. In this paper, we explore the effect of high frequency periodic insulin stimulation on Akt (also known as PKB), a crucial crosstalk node in the insulin signalling pathway that coordinates metabolic and mitogenic processes in the cell. We analyse a mathematical model of Akt translocation to the plasma membrane under both single step insulin perturbations and periodic insulin stimulation with an emphasis on - but not limited to - the physiological range of parameter values. It was shown that the system rapidly attains a robust dynamic steady state entrained to the periodic insulin stimulation. Moreover, the translocation of Akt to the plasma membrane in the model permits a sufficient level of phosphorylation to trigger downstream metabolic regulators. However, the modelling also indicated that further investigation of this activation process is required to determine whether the response of Akt is a key determinant of the enhanced metabolic control observed under periodic insulin stimulation.

Computational insight into energy control balance by Ca2+ and cAMP-PKA signaling in pacemaker cells

Cyclic adenosine monophosphate (cAMP)-protein kinase A (PKA) signaling controls sinoatrial node cell (SANC) function by affecting the degree of coupling between Ca2+ and membrane clocks. PKA is known to phosphorylate ionic channels, Ca2+ pump and release from the sarcoplasmic reticulum, and enzymes controlling ATP production in the mitochondria. While the PKA cytosolic targets in SANC have been extensively explored, its mitochondrial targets and its ability to maintain SANC energetic balance remain to be elucidated. To investigate the role of PKA in SANC energetics, we tested three hypotheses: (i) PKA is an important regulator of the ATP supply-to-demand balance, (ii) Ca2+ regulation of energetics is important for maintenance of NADH level and (iii) abrupt reduction in ATP demand first reduces the AP firing rate and, after dropping below a certain threshold, leads to a reduction in ATP. To gain mechanistic insights into the ATP supply-to-demand matching regulators, a modified model of mitochondrial energy metabolism was integrated into our coupled-clock model that describes ATP demand. Experimentally, increased ATP demand was accompanied by maintained ATP and NADH levels. Ca2+ regulation of energetics was found by the model to be important in the maintenance of NADH and PKA regulation was found to be important in the maintenance of intracellular ATP and the increase in oxygen consumption. PKA inhibition led to a biphasic reduction in AP firing rate, with the first phase being rapid and ATP-independent, while the second phase was slow and ATP-dependent. Thus, SANC energy balance is maintained by both Ca2+ and PKA signaling.

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.

Are physiological oscillations physiological?

Despite widespread and striking examples of physiological oscillations, their functional role is often unclear. Even glycolysis, the paradigm example of oscillatory biochemistry, has seen questions about its oscillatory function. Here, we take a systems approach to argue that oscillations play critical physiological roles, such as enabling systems to avoid desensitization, to avoid chronically high and therefore toxic levels of chemicals, and to become more resistant to noise. Oscillation also enables complex physiological systems to reconcile incompatible conditions such as oxidation and reduction, by cycling between them, and to synchronize the oscillations of many small units into one large effect. In pancreatic β-cells, glycolytic oscillations synchronize with calcium and mitochondrial oscillations to drive pulsatile insulin release, critical for liver regulation of glucose. In addition, oscillation can keep biological time, essential for embryonic development in promoting cell diversity and pattern formation. The functional importance of oscillatory processes requires a re-thinking of the traditional doctrine of homeostasis, holding that physiological quantities are maintained at constant equilibrium values, a view that has largely failed in the clinic. A more dynamic approach will initiate a paradigm shift in our view of health and disease. A deeper look into the mechanisms that create, sustain and abolish oscillatory processes requires the language of nonlinear dynamics, well beyond the linearization techniques of equilibrium control theory. Nonlinear dynamics enables us to identify oscillatory ('pacemaking') mechanisms at the cellular, tissue and system levels.

Spatial distribution of heterogeneity as a modulator of collective dynamics in pancreatic beta-cell networks and beyond

We study the impact of spatial distribution of heterogeneity on collective dynamics in gap-junction coupled beta-cell networks comprised on cells from two populations that differ in their intrinsic excitability. Initially, these populations are uniformly and randomly distributed throughout the networks. We develop and apply an iterative algorithm for perturbing the arrangement of the network such that cells from the same population are increasingly likely to be adjacent to one another. We find that the global input strength, or network drive, necessary to transition the network from a state of quiescence to a state of synchronised and oscillatory activity decreases as network sortedness increases. Moreover, for weak coupling, we find that regimes of partial synchronisation and wave propagation arise, which depend both on network drive and network sortedness. We then demonstrate the utility of this algorithm for studying the distribution of heterogeneity in general networks, for which we use Watts-Strogatz networks as a case study. This work highlights the importance of heterogeneity in node dynamics in establishing collective rhythms in complex, excitable networks and has implications for a wide range of real-world systems that exhibit such heterogeneity.

The Dynamics of Calcium Signaling in Beta Cells-A Discussion on the Comparison of Experimental and Modelling Data

The stimulus-secretion coupling of the pancreatic beta cell is particularly complex, as it integrates the availability of glucose and other nutrients with the neuronal and hormonal input to generate rates of insulin secretion that are appropriate for the entire organism. It is beyond dispute however, that the cytosolic Ca²⁺ concentration plays a particularly prominent role in this process, as it not only triggers the fusion of insulin granules with the plasma membrane, but also regulates the metabolism of nutrient secretagogues and affects the function of ion channels and transporters. In order to obtain a better understanding of the interdependence of these processes and, ultimately, of the entire beta cell as a working system, models have been developed based on a set of nonlinear ordinary differential equations, and were tested and parametrized on a limited set of experiments. In the present investigation, we have used a recently published version of the beta cell model to test its ability to describe further measurements from our own experimentation and from the literature. The sensitivity of the parameters is quantified and discussed; furthermore, the possible influence of the measuring technique is taken into account. The model proved to be powerful in correctly describing the depolarization pattern in response to glucose and the reaction of the cytosolic Ca²⁺ concentration to stepwise increases of the extracellular K⁺ concentration. Additionally, the membrane potential during a KATP channel block combined with a high extracellular K⁺ concentration could be reproduced. In some cases, however, a slight change of a single parameter led to an abrupt change in the cellular response, such as the generation of a Ca²⁺ oscillation with high amplitude and high frequency. This raises the question as to whether the beta cell may be a partially unstable system or whether further developments in modeling are needed to achieve a generally valid description of the stimulus-secretion coupling of the beta cell.

Metabolic cycles and signals for insulin secretion

In this review, we focus on recent developments in our understanding of nutrient-induced insulin secretion that challenge a key aspect of the "canonical" model, in which an oxidative phosphorylation-driven rise in ATP production closes K_ATP channels. We discuss the importance of intrinsic β cell metabolic oscillations; the phasic alignment of relevant metabolic cycles, shuttles, and shunts; and how their temporal and compartmental relationships align with the triggering phase or the secretory phase of pulsatile insulin secretion. Metabolic signaling components are assigned regulatory, effectory, and/or homeostatic roles vis-à-vis their contribution to glucose sensing, signal transmission, and resetting the system. Taken together, these functions provide a framework for understanding how allostery, anaplerosis, and oxidative metabolism are integrated into the oscillatory behavior of the secretory pathway. By incorporating these temporal as well as newly discovered spatial aspects of β cell metabolism, we propose a much-refined Mito_Cat-Mito_Ox model of the signaling process for the field to evaluate.

Pulsatile Basal Insulin Secretion Is Driven by Glycolytic Oscillations

In fasted and fed states, blood insulin levels are oscillatory. While this phenomenon is well studied at high glucose levels, comparatively little is known about its origin under basal conditions. We propose a possible mechanism for basal insulin oscillations based on oscillations in glycolysis, demonstrated using an established mathematical model. At high glucose, this is superseded by a calcium-dependent mechanism.

Contribution of Mitochondria to Insulin Secretion by Various Secretagogues

Significance: Mitochondria determine glucose-stimulated insulin secretion (GSIS) in pancreatic β-cells by elevating ATP synthesis. As the metabolic and redox hub, mitochondria provide numerous links to the plasma membrane channels, insulin granule vesicles (IGVs), cell redox, NADH, NADPH, and Ca²⁺ homeostasis, all affecting insulin secretion. Recent Advances: Mitochondrial redox signaling was implicated in several modes of insulin secretion (branched-chain ketoacid [BCKA]-, fatty acid [FA]-stimulated). Mitochondrial Ca²⁺ influx was found to enhance GSIS, reflecting cytosolic Ca²⁺ oscillations induced by action potential spikes (intermittent opening of voltage-dependent Ca²⁺ and K⁺ channels) or the superimposed Ca²⁺ release from the endoplasmic reticulum (ER). The ATPase inhibitory factor 1 (IF1) was reported to tune the glucose sensitivity range for GSIS. Mitochondrial protein kinase A was implicated in preventing the IF1-mediated inhibition of the ATP synthase. Critical Issues: It is unknown how the redox signal spreads up to the plasma membrane and what its targets are, what the differences in metabolic, redox, NADH/NADPH, and Ca²⁺ signaling, and homeostasis are between the first and second GSIS phase, and whether mitochondria can replace ER in the amplification of IGV exocytosis. Future Directions: Metabolomics studies performed to distinguish between the mitochondrial matrix and cytosolic metabolites will elucidate further details. Identifying the targets of cell signaling into mitochondria and of mitochondrial retrograde metabolic and redox signals to the cell will uncover further molecular mechanisms for insulin secretion stimulated by glucose, BCKAs, and FAs, and the amplification of secretion by glucagon-like peptide (GLP-1) and metabotropic receptors. They will identify the distinction between the hub β-cells and their followers in intact and diabetic states. Antioxid. Redox Signal. 36, 920-952.

Slow oscillations persist in pancreatic beta cells lacking phosphofructokinase M

Pulsatile insulin secretion by pancreatic beta cells is necessary for tight glucose control in the body. Glycolytic oscillations have been proposed as the mechanism for generating the electrical oscillations underlying pulsatile insulin secretion. The glycolytic enzyme 6-phosphofructokinase-1 (PFK) synthesizes fructose-1,6-bisphosphate (FBP) from fructose-6-phosphate. It has been proposed that the slow electrical and Ca2+ oscillations (periods of 3-5 min) observed in islets result from allosteric feedback activation of PFKM by FBP. Pancreatic beta cells express three PFK isozymes: PFKL, PFKM, and PFKP. A prior study of mice that were engineered to lack PFKM using a gene-trap strategy to delete Pfkm produced a mosaic reduction in global Pfkm expression, but the islets isolated from the mice still exhibited slow Ca2+ oscillations. However, these islets still expressed residual PFKM protein. Thus, to more fully test the hypothesis that beta cell PFKM is responsible for slow islet oscillations, we made a beta-cell-specific knockout mouse that completely lacked PFKM. While PFKM deletion resulted in subtle metabolic changes in vivo, islets that were isolated from these mice continued to exhibit slow oscillations in electrical activity, beta cell Ca2+ concentrations, and glycolysis, as measured using PKAR, an FBP reporter/biosensor. Furthermore, simulations obtained with a mathematical model of beta cell activity shows that slow oscillations can persist despite PFKM loss provided that one of the other PFK isoforms, such as PFKP, is present, even if its level of expression is unchanged. Thus, while we believe that PFKM may be the main regulator of slow oscillations in wild-type islets, PFKP can provide functional redundancy. Our model also suggests that PFKM likely dominates, in vivo, because it outcompetes PFKP with its higher FBP affinity and lower ATP affinity. We thus propose that isoform redundancy may rescue key physiological processes of the beta cell in the absence of certain critical genes.

Symbiosis of Electrical and Metabolic Oscillations in Pancreatic β-Cells

Insulin is secreted in a pulsatile pattern, with important physiological ramifications. In pancreatic β-cells, which are the cells that synthesize insulin, insulin exocytosis is elicited by pulses of elevated intracellular Ca²⁺ initiated by bursts of electrical activity. In parallel with these electrical and Ca²⁺ oscillations are oscillations in metabolism, and the periods of all of these oscillatory processes are similar. A key question that remains unresolved is whether the electrical oscillations are responsible for the metabolic oscillations via the effects of Ca²⁺, or whether the metabolic oscillations are responsible for the electrical oscillations due to the effects of ATP on ATP-sensitive ion channels? Mathematical modeling is a useful tool for addressing this and related questions as modeling can aid in the design of well-focused experiments that can test the predictions of particular models and subsequently be used to improve the models in an iterative fashion. In this article, we discuss a recent mathematical model, the Integrated Oscillator Model (IOM), that was the product of many years of development. We use the model to demonstrate that the relationship between calcium and metabolism in beta cells is symbiotic: in some contexts, the electrical oscillations drive the metabolic oscillations, while in other contexts it is the opposite. We provide new insights regarding these results and illustrate that what might at first appear to be contradictory data are actually compatible when viewed holistically with the IOM.

Measurement of Pulsatile Insulin Secretion: Rationale and Methodology

Pancreatic β-cells are responsible for the synthesis and exocytosis of insulin in response to an increase in circulating glucose. Insulin secretion occurs in a pulsatile manner, with oscillatory pulses superimposed on a basal secretion rate. Insulin pulses are a marker of β-cell health, and secretory parameters, such as pulse amplitude, time interval and frequency distribution, are impaired in obesity, aging and type 2 diabetes. In this review, we detail the mechanisms of insulin production and β-cell synchronization that regulate pulsatile insulin secretion, and we discuss the challenges to consider when measuring fast oscillatory secretion in vivo. These include the anatomical difficulties of measuring portal vein insulin noninvasively in humans before the hormone is extracted by the liver and quickly removed from the circulation. Peripheral concentrations of insulin or C-peptide, a peptide cosecreted with insulin, can be used to estimate their secretion profile, but mathematical deconvolution is required. Parametric and nonparametric approaches to the deconvolution problem are evaluated, alongside the assumptions and trade-offs required for their application in the quantification of unknown insulin secretory rates from known peripheral concentrations. Finally, we discuss the therapeutical implication of targeting impaired pulsatile secretion and its diagnostic value as an early indicator of β-cell stress.

Assessing Different Temporal Scales of Calcium Dynamics in Networks of Beta Cell Populations

Beta cells within the pancreatic islets of Langerhans respond to stimulation with coherent oscillations of membrane potential and intracellular calcium concentration that presumably drive the pulsatile exocytosis of insulin. Their rhythmic activity is multimodal, resulting from networked feedback interactions of various oscillatory subsystems, such as the glycolytic, mitochondrial, and electrical/calcium components. How these oscillatory modules interact and affect the collective cellular activity, which is a prerequisite for proper hormone release, is incompletely understood. In the present work, we combined advanced confocal Ca2+ imaging in fresh mouse pancreas tissue slices with time series analysis and network science approaches to unveil the glucose-dependent characteristics of different oscillatory components on both the intra- and inter-cellular level. Our results reveal an interrelationship between the metabolically driven low-frequency component and the electrically driven high-frequency component, with the latter exhibiting the highest bursting rates around the peaks of the slow component and the lowest around the nadirs. Moreover, the activity, as well as the average synchronicity of the fast component, considerably increased with increasing stimulatory glucose concentration, whereas the stimulation level did not affect any of these parameters in the slow component domain. Remarkably, in both dynamical components, the average correlation decreased similarly with intercellular distance, which implies that intercellular communication affects the synchronicity of both types of oscillations. To explore the intra-islet synchronization patterns in more detail, we constructed functional connectivity maps. The subsequent comparison of network characteristics of different oscillatory components showed more locally clustered and segregated networks of fast oscillatory activity, while the slow oscillations were more global, resulting in several long-range connections and a more cohesive structure. Besides the structural differences, we found a relatively weak relationship between the fast and slow network layer, which suggests that different synchronization mechanisms shape the collective cellular activity in islets, a finding which has to be kept in mind in future studies employing different oscillations for constructing networks.

The Pancreatic β-Cell: The Perfect Redox System

Pancreatic β-cell insulin secretion, which responds to various secretagogues and hormonal regulations, is reviewed here, emphasizing the fundamental redox signaling by NADPH oxidase 4- (NOX4-) mediated H₂O₂ production for glucose-stimulated insulin secretion (GSIS). There is a logical summation that integrates both metabolic plus redox homeostasis because the ATP-sensitive K⁺ channel (K_ATP) can only be closed when both ATP and H₂O₂ are elevated. Otherwise ATP would block K_ATP, while H₂O₂ would activate any of the redox-sensitive nonspecific calcium channels (NSCCs), such as TRPM2. Notably, a 100%-closed K_ATP ensemble is insufficient to reach the -50 mV threshold plasma membrane depolarization required for the activation of voltage-dependent Ca²⁺ channels. Open synergic NSCCs or Cl^- channels have to act simultaneously to reach this threshold. The resulting intermittent cytosolic Ca²⁺-increases lead to the pulsatile exocytosis of insulin granule vesicles (IGVs). The incretin (e.g., GLP-1) amplification of GSIS stems from receptor signaling leading to activating the phosphorylation of TRPM channels and effects on other channels to intensify integral Ca²⁺-influx (fortified by endoplasmic reticulum Ca²⁺). ATP plus H₂O₂ are also required for branched-chain ketoacids (BCKAs); and partly for fatty acids (FAs) to secrete insulin, while BCKA or FA β-oxidation provide redox signaling from mitochondria, which proceeds by H₂O₂ diffusion or hypothetical SH relay via peroxiredoxin "redox kiss" to target proteins.

CDK2 limits the highly energetic secretory program of mature β cells by restricting PEP cycle-dependent KATP channel closure

Hallmarks of mature β cells are restricted proliferation and a highly energetic secretory state. Paradoxically, cyclin-dependent kinase 2 (CDK2) is synthesized throughout adulthood, its cytosolic localization raising the likelihood of cell cycle-independent functions. In the absence of any changes in β cell mass, maturity, or proliferation, genetic deletion of Cdk2 in adult β cells enhanced insulin secretion from isolated islets and improved glucose tolerance in vivo. At the single β cell level, CDK2 restricts insulin secretion by increasing K_ATP conductance, raising the set point for membrane depolarization in response to activation of the phosphoenolpyruvate (PEP) cycle with mitochondrial fuels. In parallel with reduced β cell recruitment, CDK2 restricts oxidative glucose metabolism while promoting glucose-dependent amplification of insulin secretion. This study provides evidence of essential, non-canonical functions of CDK2 in the secretory pathways of quiescent β cells.

Despite loss of proliferative capacity with age, mature β cells continually synthesize CDK2. Sdao et al. demonstrate that CDK2 depletion in adult β cells improves glucose tolerance in vivo. By augmenting PEP cycle-dependent K_ATP channel closure, CDK2 inactivation lowers the set point for membrane depolarization, augmenting oxidative metabolism and insulin secretion.

Phantom bursting may underlie electrical bursting in single pancreatic β-cells

Insulin is secreted by pancreatic β-cellsthat are electrically coupled into micro-organs called islets of Langerhans. The secretion is due to the influx of Ca²⁺ions that accompany electrical impulses, which are clustered into bursts. So-called "medium bursting" occurs in many β-cellsin intact islets, while in other islets the β-cellsexhibit "slow bursting", with a much longer period. Each burst brings in Ca²⁺ that, through exocytosis, results in insulin secretion. When isolated from an islet, β-cellsbehave very differently. The electrical activity is much noisier, and consists primarily of trains of irregularly-timed spikes, or fast or slow bursting. Medium bursting, so often seen in intact islets, is rarely if ever observed. In this study, we examine what the isolated cell behavior can tell us about the mechanism for bursting in intact islets. A previous mathematical study concluded that the slow bursting observed in isolated β-cells, and therefore most likely in islets, must be due to intrinsic glycolytic oscillations, since this mechanism for bursting is robust to noise. It was demonstrated that an alternate mechanism, phantom bursting, was very sensitive to noise, and therefore could not account for the slow bursting in single cells. We re-examine these conclusions, motivated by recent experimental and mathematical modeling evidence that slow bursting in intact islets is, at least in many cases, driven by the phantom bursting mechanism and not endogenous glycolytic oscillations. We employ two phantom bursting models, one minimal and the other more biophysical, to determine the sensitivity of medium and slow bursting to electrical current noise. In the minimal model, both forms of bursting are highly sensitive to noise. In the biophysical model, while medium bursting is sensitive to noise, slow bursting is much less sensitive. This suggests that the slow bursting seen in isolated β-cellsmay be due to a phantom bursting mechanism, and by extension, slow bursting in intact islets may also be driven by this mechanism.

Chronic stimulation induces adaptive potassium channel activity that restores calcium oscillations in pancreatic islets in vitro

Insulin pulsatility is important to hepatic response in regulating blood glucose. Growing evidence suggests that insulin-secreting pancreatic β-cells can adapt to chronic disruptions of pulsatility to rescue this physiologically important behavior. We determined the time scale for adaptation and examined potential ion channels underlying it. We induced the adaptation both by chronic application of the ATP-sensitive K⁺ [K(ATP)] channel blocker tolbutamide and by application of the depolarizing agent potassium chloride (KCl). Acute application of tolbutamide without pretreatment results in elevated Ca²⁺ as measured by fura-2AM and the loss of endogenous pulsatility. We show that after chronic exposure to tolbutamide (12-24 h), Ca²⁺ oscillations occur with subsequent acute tolbutamide application. The same experiment was conducted with potassium chloride (KCl) to directly depolarize the β-cells. Once again, following chronic exposure to the cell stimulator, the islets produced Ca²⁺ oscillations when subsequently exposed to tolbutamide. These experiments suggest that it is the chronic stimulation, and not tolbutamide desensitization, that is responsible for the adaptation that rescues oscillatory β-cell activity. This compensatory response also causes islet glucose sensitivity to shift rightward following chronic tolbutamide treatment. Mathematical modeling shows that a small increase in the number of K(ATP) channels in the membrane is one adaptation mechanism that is compatible with the data. To examine other compensatory mechanisms, pharmacological studies provide support that Kir2.1 and TEA-sensitive channels play some role. Overall, this investigation demonstrates β-cell adaptability to overstimulation, which is likely an important mechanism for maintaining glucose homeostasis in the face of chronic stimulation.

Metabolic regulation of calcium signaling in beta cells

The cytoplasmic Ca²⁺ concentration ([Ca²⁺]_cyt) regulates a vast number of cellular functions, including insulin secretion from beta cells. The major physiological insulin secretagogue, glucose, triggers [Ca²⁺]_cyt oscillations in beta cells. Synchronization of the oscillations among the beta cells within an islet underlies the generation of pulsatile insulin secretion. This review describes the mechanisms generating [Ca²⁺]_cyt oscillations, the interactions between [Ca²⁺]_cyt and cell metabolism, as well as the contribution of various organelles to the shaping of [Ca²⁺]_cyt signals and insulin secretion. It also discusses how Ca²⁺ signals are coordinated and spread throughout the islets and data indicating that altered Ca²⁺ signaling is associated with beta cell dysfunction and development of type 2 diabetes.

Integrated multiscale mathematical modeling of insulin secretion reveals the role of islet network integrity for proper oscillatory glucose-dose response

The integrated multiscale mathematical model we present in this paper is built on two of our previous ones: a model of electrical oscillation in β-cells connected to neighboring cells within a three-dimensional (3D) network, and a model of glucose-induced β-cell intracellular insulin granule trafficking and insulin secretion. In order to couple these two models, we assume that the rate at which primed and release-ready insulin granules fuse at the cell membrane increases with the intracellular calcium concentration. Moreover, by assuming that the fraction of free K_ATP-channels decreases with increasing glucose concentration, we take into account the effect of glucose dose on membrane potential and, indirectly via the effect on the potential, on intracellular calcium. Numerical analysis of our new model shows that a single step increase in glucose concentration yields the experimentally observed characteristic biphasic insulin release. We find that the biphasic response is typically oscillatory in nature for low and moderate glucose concentrations. The plateau fraction (the time that the β-cells spend in their active firing phase) increases with increasing glucose dose, as does the total insulin secretion. At high glucose concentrations, the oscillations tend to vanish due to a constantly elevated membrane potential of the β-cells. Our results also demonstrate how insulin secretion characteristics in various glucose protocols depend on the degree of β-cell loss, highlighting the potential impact from disease. In particular, both the secretory capacity (average insulin secretion rate per β-cell) and the oscillatory response diminish as the islet cell network becomes compromised.

Mathematical Modelling of Endocrine Systems

Hormone rhythms are ubiquitous and essential to sustain normal physiological functions. Combined mathematical modelling and experimental approaches have shown that these rhythms result from regulatory processes occurring at multiple levels of organisation and require continuous dynamic equilibration, particularly in response to stimuli. We review how such an interdisciplinary approach has been successfully applied to unravel complex regulatory mechanisms in the metabolic, stress, and reproductive axes. We discuss how this strategy is likely to be instrumental for making progress in emerging areas such as chronobiology and network physiology. Ultimately, we envisage that the insight provided by mathematical models could lead to novel experimental tools able to continuously adapt parameters to gradual physiological changes and the design of clinical interventions to restore normal endocrine function.

Nutritional recovery from a low-protein diet during pregnancy does not restore the kinetics of insulin secretion and Ca2+ or alterations in the cAMP/PKA and PLC/PKC pathways in islets from adult rats

We investigated the insulin release induced by glucose, the Ca²⁺ oscillatory pattern, and the cyclic AMP (cAMP)/protein kinase A (PKA) and phospholipase C (PLC)/protein kinase C (PKC) pathways in islets from adult rats that were reared under diets with 17% protein (C) or 6% protein (LP) during gestation, suckling, and after weaning and in rats receiving diets with 6% protein during gestation and 17% protein after birth (R). First-phase glucose-induced insulin secretion was reduced in LP and R islets, and the second phase was partially restored in the R group. Glucose stimulation did not modify intracellular Ca²⁺ concentration, but it reduced the Ca²⁺ oscillatory frequency in the R group compared with the C group. Intracellular cAMP concentration was higher and PKA-Cα expression was lower in the R and LP groups compared with the C group. The PKCα content in islets from R rats was lower than that in C and LP rats. Thus, nutritional recovery from a low-protein diet during fetal life did not repair the kinetics of insulin release, impaired Ca²⁺ handling, and altered the cAMP/PKA and PLC/PKC pathways.

Transitions between bursting modes in the integrated oscillator model for pancreatic β-cells

Insulin-secreting β-cells of pancreatic islets of Langerhans produce bursts of electrical impulses, resulting in intracellular Ca²⁺ oscillations and pulsatile insulin secretion. The mechanism for this bursting activity has been the focus of mathematical modeling for more than three decades, and as new data are acquired old models are modified and new models are developed. Comprehensive models must now account for the various modes of bursting observed in islet β-cells, which include fast bursting, slow bursting, and compound bursting. One such model is the Integrated Oscillator Model (IOM), in which β-cell electrical activity, intracellular Ca²⁺, and glucose metabolism interact via numerous feedforward and feedback pathways. These interactions can produce metabolic oscillations with a sawtooth time course or a pulsatile time course, reflecting very different oscillation mechanisms. In this report, we determine conditions favorable to one form of oscillations or the other, and examine the transitions between modes of bursting and the relationship of the transitions to the patterns of metabolic oscillations. Importantly, this work clarifies what can be expected in experimental measurements of β-cell oscillatory activity, and suggests pathways through which oscillations of one type can be converted to oscillations of another type.

Closing in on the Mechanisms of Pulsatile Insulin Secretion

Insulin secretion from pancreatic islet β-cells occurs in a pulsatile fashion, with a typical period of ∼5 min. The basis of this pulsatility in mouse islets has been investigated for more than four decades, and the various theories have been described as either qualitative or mathematical models. In many cases the models differ in their mechanisms for rhythmogenesis, as well as other less important details. In this Perspective, we describe two main classes of models: those in which oscillations in the intracellular Ca²⁺ concentration drive oscillations in metabolism, and those in which intrinsic metabolic oscillations drive oscillations in Ca²⁺ concentration and electrical activity. We then discuss nine canonical experimental findings that provide key insights into the mechanism of islet oscillations and list the models that can account for each finding. Finally, we describe a new model that integrates features from multiple earlier models and is thus called the Integrated Oscillator Model. In this model, intracellular Ca²⁺ acts on the glycolytic pathway in the generation of oscillations, and it is thus a hybrid of the two main classes of models. It alone among models proposed to date can explain all nine key experimental findings, and it serves as a good starting point for future studies of pulsatile insulin secretion from human islets.

Regulation of ATP utilization during metastatic cell migration by collagen architecture

Cell migration in a three-dimensional matrix requires that cells either remodel the surrounding matrix fibers and/or squeeze between the fibers to move. Matrix degradation, matrix remodeling, and changes in cell shape each require cells to expend energy. While significant research has been performed to understand the cellular and molecular mechanisms guiding metastatic migration, less is known about cellular energy regulation and utilization during three-dimensional cancer cell migration. Here we introduce the use of the genetically encoded fluorescent biomarkers, PercevalHR and pHRed, to quantitatively assess ATP, ADP, and pH levels in MDA-MB-231 metastatic cancer cells as a function of the local collagen microenvironment. We find that the use of the probe is an effective tool for exploring the thermodynamics of cancer cell migration and invasion. Specifically, we find that the ATP:ADP ratio increases in cells in denser matrices, where migration is impaired, and it decreases in cells in aligned collagen matrices, where migration is facilitated. When migration is pharmacologically inhibited, the ATP:ADP ratio decreases. Together, our data indicate that matrix architecture alters cellular energetics and that intracellular ATP:ADP ratio is related to the ability of cancer cells to effectively migrate.

Local and regional control of calcium dynamics in the pancreatic islet

Ca²⁺ is the key intracellular regulator of insulin secretion, acting in the β-cell as the ultimate trigger for exocytosis. In response to high glucose, ATP-sensitive K⁺ channel closure and plasma membrane depolarization engage a sophisticated machinery to drive pulsatile cytosolic Ca²⁺ changes. Voltage-gated Ca²⁺ channels, Ca²⁺ -activated K⁺ channels and Na⁺ /Ca²⁺ exchange all play important roles. The use of targeted Ca²⁺ probes has revealed that during each cytosolic Ca²⁺ pulse, uptake of Ca²⁺ by mitochondria, endoplasmic reticulum (ER), secretory granules and lysosomes fine-tune cytosolic Ca²⁺ dynamics and control organellar function. For example, changes in the expression of the Ca²⁺ -binding protein Sorcin appear to provide a link between ER Ca²⁺ levels and ER stress, affecting β-cell function and survival. Across the islet, intercellular communication between highly interconnected "hubs," which act as pacemaker β-cells, and subservient "followers," ensures efficient insulin secretion. Loss of connectivity is seen after the deletion of genes associated with type 2 diabetes (T2D) and follows metabolic and inflammatory insults that characterize this disease. Hubs, which typically comprise ~1%-10% of total β-cells, are repurposed for their specialized role by expression of high glucokinase (Gck) but lower Pdx1 and Nkx6.1 levels. Single cell-omics are poised to provide a deeper understanding of the nature of these cells and of the networks through which they communicate. New insights into the control of both the intra- and intercellular Ca²⁺ dynamics may thus shed light on T2D pathology and provide novel opportunities for therapy.

Upregulation of an inward rectifying K+ channel can rescue slow Ca2+ oscillations in K(ATP) channel deficient pancreatic islets

Plasma insulin oscillations are known to have physiological importance in the regulation of blood glucose. In insulin-secreting β-cells of pancreatic islets, K(ATP) channels play a key role in regulating glucose-dependent insulin secretion. In addition, they convey oscillations in cellular metabolism to the membrane by sensing adenine nucleotides, and are thus instrumental in mediating pulsatile insulin secretion. Blocking K(ATP) channels pharmacologically depolarizes the β-cell plasma membrane and terminates islet oscillations. Surprisingly, when K(ATP) channels are genetically knocked out, oscillations in islet activity persist, and relatively normal blood glucose levels are maintained. Compensation must therefore occur to overcome the loss of K(ATP) channels in K(ATP) knockout mice. In a companion study, we demonstrated a substantial increase in Kir2.1 protein occurs in β-cells lacking K(ATP) because of SUR1 deletion. In this report, we demonstrate that β-cells of SUR1 null islets have an upregulated inward rectifying K⁺ current that helps to compensate for the loss of K(ATP) channels. This current is likely due to the increased expression of Kir2.1 channels. We used mathematical modeling to determine whether an ionic current having the biophysical characteristics of Kir2.1 is capable of rescuing oscillations that are similar in period to those of wild-type islets. By experimentally testing a key model prediction we suggest that Kir2.1 current upregulation is a likely mechanism for rescuing the oscillations seen in islets from mice deficient in K(ATP) channels.

Pulsatile insulin secretion is important for the proper regulation of blood glucose, and disruption of this pulsatility is a hallmark of type II diabetes. An ion channel was discovered more than three decades ago that conveys the metabolic state of insulin-secreting β-cells to the plasma membrane because it is blocked by ATP and opened by ADP, and thereby controls the activity of these electrically-excitable cells on a rapid time scale according to the prevailing blood glucose level. In addition to setting the appropriate level of insulin secretion, K(ATP) channels play a key role in generating the oscillations in cellular activity that underlie insulin pulsatility. It is therefore surprising that oscillations in activity persist in islets in which the K(ATP) channels are genetically knocked out. In this combined modeling and experimental study, we demonstrate that the role played by K(ATP) current in wild-type β-cells can be taken over by an inward-rectifying K⁺ current which, we show here, is upregulated in β-cells from SUR1 knockout mice. This result helps to resolve a mystery in the field that has remained elusive for more than a decade, since the first studies showing oscillations in SUR1^-/- islets.

Mechanisms of the amplifying pathway of insulin secretion in the β cell

Pancreatic islet β cells secrete insulin in response to nutrient secretagogues, like glucose, dependent on calcium influx and nutrient metabolism. One of the most intriguing qualities of β cells is their ability to use metabolism to amplify the amount of secreted insulin independent of further alterations in intracellular calcium. Many years studying this amplifying process have shaped our current understanding of β cell stimulus-secretion coupling; yet, the exact mechanisms of amplification have been elusive. Recent studies utilizing metabolomics, computational modeling, and animal models have progressed our understanding of the metabolic amplifying pathway of insulin secretion from the β cell. New approaches will be discussed which offer in-roads to a more complete model of β cell function. The development of β cell therapeutics may be aided by such a model, facilitating the targeting of aspects of the metabolic amplifying pathway which are unique to the β cell.

Calcium Oscillation Frequency-Sensitive Gene Regulation and Homeostatic Compensation in Pancreatic β-Cells

Pancreatic islet [Formula: see text]-cells are electrically excitable cells that secrete insulin in an oscillatory fashion when the blood glucose concentration is at a stimulatory level. Insulin oscillations are the result of cytosolic [Formula: see text] oscillations that accompany bursting electrical activity of [Formula: see text]-cells and are physiologically important. ATP-sensitive [Formula: see text] channels (K(ATP) channels) play the key role in setting the overall activity of the cell and in driving bursting, by coupling cell metabolism to the membrane potential. In humans, when there is a defect in K(ATP) channel function, [Formula: see text]-cells fail to respond appropriately to changes in the blood glucose level, and electrical and [Formula: see text] oscillations are lost. However, mice compensate for K(ATP) channel defects in islet [Formula: see text]-cells by employing alternative mechanisms to maintain electrical and [Formula: see text] oscillations. In a recent study, we showed that in mice islets in which K(ATP) channels are genetically knocked out another [Formula: see text] current, provided by inward-rectifying [Formula: see text] channels, is increased. With mathematical modeling, we demonstrated that a sufficient upregulation in these channels can account for the paradoxical electrical bursting and [Formula: see text] oscillations observed in these [Formula: see text]-cells. However, the question of determining the correct level of upregulation that is necessary for this compensation remained unanswered, and this question motivates the current study. [Formula: see text] is a well-known regulator of gene expression, and several examples have been shown of genes that are sensitive to the frequency of the [Formula: see text] signal. In this mathematical modeling study, we demonstrate that a [Formula: see text] oscillation frequency-sensitive gene transcription network can adjust the gene expression level of a compensating [Formula: see text] channel so as to rescue electrical bursting and [Formula: see text] oscillations in a model [Formula: see text]-cell in which the key K(ATP) current is removed. This is done without the prescription of a target [Formula: see text] level, but evolves naturally as a consequence of the feedback between the [Formula: see text]-dependent enzymes and the cell's electrical activity. More generally, the study indicates how [Formula: see text] can provide the link between gene expression and cellular electrical activity that promotes wild-type behavior in a cell following gene knockout.

Phase Analysis of Metabolic Oscillations and Membrane Potential in Pancreatic Islet β-Cells

Metabolism in islet β-cells displays oscillations that can trigger pulses of electrical activity and insulin secretion. There has been a decades-long debate among islet biologists about whether metabolic oscillations are intrinsic or occur in response to oscillations in intracellular Ca(2+) that result from bursting electrical activity. In this article, the dynamics of oscillatory metabolism were investigated using five different optical reporters. Reporter activity was measured simultaneously with membrane potential bursting to determine the phase relationships between the metabolic oscillations and electrical activity. Our experimental findings suggest that Ca(2+) entry into β-cells stimulates the rate of mitochondrial metabolism, accounting for the depletion of glycolytic intermediates during each oscillatory burst. We also performed Ca(2+) clamp tests in which we clamped membrane potential with the KATP channel-opener diazoxide and KCl to fix Ca(2+) at an elevated level. These tests confirm that metabolic oscillations do not require Ca(2+) oscillations, but show that Ca(2+) plays a larger role in shaping metabolic oscillations than previously suspected. A dynamical picture of the mechanisms of oscillations emerged that requires the restructuring of contemporary mathematical β-cell models, including our own dual oscillator model. In the companion article, we modified our model to account for these new data.

Glucose Oscillations Can Activate an Endogenous Oscillator in Pancreatic Islets

Pancreatic islets manage elevations in blood glucose level by secreting insulin into the bloodstream in a pulsatile manner. Pulsatile insulin secretion is governed by islet oscillations such as bursting electrical activity and periodic Ca²⁺ entry in β-cells. In this report, we demonstrate that although islet oscillations are lost by fixing a glucose stimulus at a high concentration, they may be recovered by subsequently converting the glucose stimulus to a sinusoidal wave. We predict with mathematical modeling that the sinusoidal glucose signal’s ability to recover islet oscillations depends on its amplitude and period, and we confirm our predictions by conducting experiments with islets using a microfluidics platform. Our results suggest a mechanism whereby oscillatory blood glucose levels recruit non-oscillating islets to enhance pulsatile insulin output from the pancreas. Our results also provide support for the main hypothesis of the Dual Oscillator Model, that a glycolytic oscillator endogenous to islet β-cells drives pulsatile insulin secretion.

A global shift throughout the last century toward excessive nutrient intake relative to energy expenditure has fueled a dramatic increase in the incidence of diabetes in humans. The epidemic is primarily of type 2 diabetes, a disease characterized by the inability of the body to effectively control blood glucose levels. Insulin plays a key role in regulating blood glucose levels by restraining endogenous glucose output from the liver and promoting blood glucose uptake by tissue throughout the body. It is secreted in pulses by islets of Langerhans, endocrine cell aggregates dispersed throughout the pancreas. Loss of insulin pulsatility is an early event in the development of type 2 diabetes. Here, we demonstrate, with a combined modeling and experimental approach, that the loss of pulsatile insulin release that results from elevated glucose may be recovered by an oscillatory glucose stimulus. Our results have potential implications for enhancing insulin pulsatility and therefore mitigating the development of type 2 diabetes.

Multi-timescale systems and fast-slow analysis

Mathematical models of biological systems often have components that vary on different timescales. This multi-timescale character can lead to problems when doing computer simulations, which can require a great deal of computer time so that the components that change on the fastest time scale can be resolved. Mathematical analysis of these multi-timescale systems can be greatly simplified by partitioning them into subsystems that evolve on different time scales. The subsystems are then analyzed semi-independently, using a technique called fast-slow analysis. In this review we describe the fast-slow analysis technique and apply it to relaxation oscillations, neuronal bursting oscillations, canard oscillations, and mixed-mode oscillations. Although these examples all involve neural systems, the technique can and has been applied to other biological, chemical, and physical systems. It is a powerful analysis method that will become even more useful in the future as new experimental techniques push forward the complexity of biological models.