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

Exploring pancreatic beta-cell subgroups and their connectivity

Functional pancreatic islet beta cells are essential to ensure glucose homeostasis across species from zebrafish to humans. These cells show significant heterogeneity, and emerging studies have revealed that connectivity across a hierarchical network is required for normal insulin release. Here, we discuss current thinking and areas of debate around intra-islet connectivity, cellular hierarchies and potential "controlling" beta-cell populations. We focus on methodologies, including comparisons of different cell preparations as well as in vitro and in vivo approaches to imaging and controlling the activity of human and rodent islet preparations. We also discuss the analytical approaches that can be applied to live-cell data to identify and study critical subgroups of cells with a disproportionate role in control Ca2+ dynamics and thus insulin secretion (such as "first responders", "leaders" and "hubs", as defined by Ca2+ responses to glucose stimulation). Possible mechanisms by which this hierarchy is achieved, its physiological relevance and how its loss may contribute to islet failure in diabetes mellitus are also considered. A glossary of terms and links to computational resources are provided.

Unveiling islet heterogeneity using an automated microfluidic imaging system

Islets of Langerhans are a therapeutic target for diabetes and prediabetes. Measurements of therapeutic inhibitory or excitatory concentrations are often performed using large groups of islets, however, the population heterogeneity cannot be observed when examining the ensemble response. Normal islet function and islet response to therapeutic treatment can be affected by islet heterogeneity, influencing the progression of diabetes mellitus. To identify heterogeneity in an islet population, we developed a simple microfluidic device capable of delivering a stimulant to four independent chambers, allowing measurements of individual responses from a population of 20-25 islets. The device enabled accurate delivery of the same stimulant concentration to all four chambers, with an error <1% between chambers. To demonstrate the capability of this system, ensemble and individual EC/IC[Formula: see text] measurements of glucose and diazoxide were performed on murine islets. Results showed little heterogeneity of glucose EC[Formula: see text] values with all 21 islets within ± 0.6 mM of the ensemble value of 7.4 mM. In contrast, application of diazoxide to 24 islets in the presence of 20 mM glucose resulted in 37% of islets having an IC[Formula: see text] greater than 10% from the ensemble value of 10.2 μM. The simple system developed here is amenable to further studies on islet heterogeneity, and is applicable to investigate heterogeneity in other cell types.

A mechanism for slow rhythms in coordinated pancreatic islet activity

Insulin levels in the blood oscillate with a variety of periods, including rapid (5-10 min), ultradian (50-120 min), and circadian (24 h). Oscillations of insulin are beneficial for lowering blood glucose and disrupted rhythms are found in people with type 2 diabetes and their close relatives. These in vivo secretion dynamics imply that the oscillatory activity of individual islets of Langerhans are synchronized, although the mechanism for this is not known. One mechanism by which islets may synchronize is negative feedback of insulin on whole-body glucose levels. In previous work, we demonstrated that a negative feedback loop with a small time delay, to account for the time required for islets to be exposed to a new glucose concentration in vivo, results in small 3-6 islet populations synchronizing to produce fast closed-loop oscillations. However, these same islet populations could also produce slow closed-loop oscillations with periods longer than the natural islet oscillation periods. Here, we investigate the origin of the slow oscillations and the bistability with the fast oscillations using larger islet populations (20-50 islets). In contrast to what was observed earlier, larger islet populations mainly synchronize to longer-period oscillations that are approximately twice the delay time used in the feedback loop. A mean-field model was also used as a proxy for a large islet population to uncover the underlying mechanism for the slow rhythm. The heterogeneous intrinsic oscillation periods of the islets interferes with this rhythm mechanism when islet populations are small, and is similar to adding noise to the mean-field model. Thus, the effect of a time delay in the glucose feedback mechanism is similar to other examples of time-delayed systems in biology and may be a viable mechanism for ultradian oscillations.

Network representation of multicellular activity in pancreatic islets: Technical considerations for functional connectivity analysis

Within the islets of Langerhans, beta cells orchestrate synchronized insulin secretion, a pivotal aspect of metabolic homeostasis. Despite the inherent heterogeneity and multimodal activity of individual cells, intercellular coupling acts as a homogenizing force, enabling coordinated responses through the propagation of intercellular waves. Disruptions in this coordination are implicated in irregular insulin secretion, a hallmark of diabetes. Recently, innovative approaches, such as integrating multicellular calcium imaging with network analysis, have emerged for a quantitative assessment of the cellular activity in islets. However, different groups use distinct experimental preparations, microscopic techniques, apply different methods to process the measured signals and use various methods to derive functional connectivity patterns. This makes comparisons between findings and their integration into a bigger picture difficult and has led to disputes in functional connectivity interpretations. To address these issues, we present here a systematic analysis of how different approaches influence the network representation of islet activity. Our findings show that the choice of methods used to construct networks is not crucial, although care is needed when combining data from different islets. Conversely, the conclusions drawn from network analysis can be heavily affected by the pre-processing of the time series, the type of the oscillatory component in the signals, and by the experimental preparation. Our tutorial-like investigation aims to resolve interpretational issues, reconcile conflicting views, advance functional implications, and encourage researchers to adopt connectivity analysis. As we conclude, we outline challenges for future research, emphasizing the broader applicability of our conclusions to other tissues exhibiting complex multicellular dynamics.

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.

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.

β-cell intrinsic dynamics rather than gap junction structure dictates subpopulations in the islet functional network

Diabetes is caused by the inability of electrically coupled, functionally heterogeneous β-cells within the pancreatic islet to provide adequate insulin secretion. Functional networks have been used to represent synchronized oscillatory [Ca2+] dynamics and to study β-cell subpopulations, which play an important role in driving islet function. The mechanism by which highly synchronized β-cell subpopulations drive islet function is unclear. We used experimental and computational techniques to investigate the relationship between functional networks, structural (gap junction) networks, and intrinsic β-cell dynamics in slow and fast oscillating islets. Highly synchronized subpopulations in the functional network were differentiated by intrinsic dynamics, including metabolic activity and KATP channel conductance, more than structural coupling. Consistent with this, intrinsic dynamics were more predictive of high synchronization in the islet functional network as compared to high levels of structural coupling. Finally, dysfunction of gap junctions, which can occur in diabetes, caused decreases in the efficiency and clustering of the functional network. These results indicate that intrinsic dynamics rather than structure drive connections in the functional network and highly synchronized subpopulations, but gap junctions are still essential for overall network efficiency. These findings deepen our interpretation of functional networks and the formation of functional subpopulations in dynamic tissues such as the islet.

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.

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.

Coordination of pancreatic islet rhythmic activity by delayed negative feedback

Secretion of insulin from the pancreas is pulsatile, driven by intrinsic oscillations within individual islets of Langerhans. The secretions are coordinated among the many islets distributed throughout the pancreas producing a synchronized rhythm in vivo that is essential for maintaining normal glucose levels. One hypothesized mechanism for the coordination of islet activity is negative feedback, whereby sequestration of glucose in response to elevated insulin leads to a reduction in the blood glucose level that is sensed by the islet population. This global signal of glucose then coordinates the individual islets. In this study, we tested how this coordination mechanism is affected by time delays in the negative feedback, using a microfluidic system to monitor Ca2+ levels in a small population of islets and implementing glucose control through a negative feedback system. We found that islet synchronization occurs even with time delays in the feedback of up to 7 min. We also found that a second, slower closed-loop oscillation period is produced during delayed feedback in which islet oscillations are clustered into episodes. The period of this second oscillatory mode increases with the time delay and appears to be a second stable behavior that coexists with the faster synchronized oscillation. The general conclusion is that islet coordination through negative feedback is a viable means of islet coordination that is robust to delays in the timing of the feedback, and could complement other potential coordination mechanisms such as entrainment by pancreatic ganglia.NEW & NOTEWORTHY Insulin secretion from islets of Langerhans is rhythmic, and these rhythms are coordinated to produce oscillatory plasma insulin levels. Using a combination of microfluidics and computational modeling, we demonstrate that coordination can occur through negative feedback of the type provided by the liver, even if that feedback is delayed by several minutes. We also demonstrate that a second, slower, mode of oscillations can occur when feedback is delayed where faster oscillations are grouped into episodes.

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.

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.

Flipping the switch on the hub cell: Islet desynchronization through cell silencing

Pancreatic β cells, responsible for secreting insulin into the bloodstream and maintaining glucose homeostasis, are organized in the islets of Langerhans as clusters of electrically coupled cells. Gap junctions, connecting neighboring cells, coordinate the behavior of the islet, leading to the synchronized oscillations in the intracellular calcium and insulin secretion in healthy islets. Recent experimental work has shown that silencing special hub cells can lead to a disruption in the coordinated behavior, calling into question the democratic paradigm of islet insulin secretion with more or less equal input from each β cell. Islets were shown to have scale-free functional connectivity and a hub cell whose silencing would lead to a loss of functional connectivity and activity in the islet. A mechanistic model representing the electrical and calcium dynamics of β cells during insulin secretion was applied to a network of cells connected by gap junctions to test the hypothesis of hub cells. Functional connectivity networks were built from the simulated calcium traces, with some networks classified as scale-free, confirming experimental results. Potential hub cells were identified using previously defined centrality measures, but silencing them was unable to desynchronize the islet. Instead, switch cells, which were able to turn off the activity of the islet but were not highly functionally connected, were found via systematically silencing each cell in the network.

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.

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.