Insulin granule trafficking in beta-cells: mathematical model of glucose-induced insulin secretion

Glucose Transporters Are Key Components of the Human Glucostat

Mouse models are extensively used in metabolic studies. However, inherent differences between the species, notably their blood glucose levels, hampered data translation into clinical settings. In this study, we confirmed GLUT1 to be the predominantly expressed glucose transporter in both adult and fetal human β-cells. In comparison, GLUT2 is detected in a small yet significant subpopulation of adult β-cells and is expressed to a greater extent in fetal β-cells. Notably, GLUT1/2 expression in INS+ cells from human stem cell-derived islet-like clusters (SC-islets) exhibited a closer resemblance to that observed in fetal islets. Transplantation of primary human islets or SC-islets, but not murine islets, lowered murine blood glucose to the human glycemic range, emphasizing the critical role of β-cells in establishing species-specific glycemia. We further demonstrate the functional requirements of GLUT1 and GLUT2 in glucose uptake and insulin secretion through chemically inhibiting GLUT1 in primary islets and SC-islets and genetically disrupting GLUT2 in SC-islets. Finally, we developed a mathematical model to predict changes in glucose uptake and insulin secretion as a function of GLUT1/2 expression. Collectively, our findings illustrate the crucial roles of GLUTs in human β-cells, and identify them as key components in establishing species-specific glycemic set points.

ARTICLE HIGHLIGHTS

A pathway model of glucose-stimulated insulin secretion in the pancreatic β-cell

The pancreas plays a critical role in maintaining glucose homeostasis through the secretion of hormones from the islets of Langerhans. Glucose-stimulated insulin secretion (GSIS) by the pancreatic β-cell is the main mechanism for reducing elevated plasma glucose. Here we present a systematic modeling workflow for the development of kinetic pathway models using the Systems Biology Markup Language (SBML). Steps include retrieval of information from databases, curation of experimental and clinical data for model calibration and validation, integration of heterogeneous data including absolute and relative measurements, unit normalization, data normalization, and model annotation. An important factor was the reproducibility and exchangeability of the model, which allowed the use of various existing tools. The workflow was applied to construct a novel data-driven kinetic model of GSIS in the pancreatic β-cell based on experimental and clinical data from 39 studies spanning 50 years of pancreatic, islet, and β-cell research in humans, rats, mice, and cell lines. The model consists of detailed glycolysis and phenomenological equations for insulin secretion coupled to cellular energy state, ATP dynamics and (ATP/ADP ratio). Key findings of our work are that in GSIS there is a glucose-dependent increase in almost all intermediates of glycolysis. This increase in glycolytic metabolites is accompanied by an increase in energy metabolites, especially ATP and NADH. One of the few decreasing metabolites is ADP, which, in combination with the increase in ATP, results in a large increase in ATP/ADP ratios in the β-cell with increasing glucose. Insulin secretion is dependent on ATP/ADP, resulting in glucose-stimulated insulin secretion. The observed glucose-dependent increase in glycolytic intermediates and the resulting change in ATP/ADP ratios and insulin secretion is a robust phenomenon observed across data sets, experimental systems and species. Model predictions of the glucose-dependent response of glycolytic intermediates and biphasic insulin secretion are in good agreement with experimental measurements. Our model predicts that factors affecting ATP consumption, ATP formation, hexokinase, phosphofructokinase, and ATP/ADP-dependent insulin secretion have a major effect on GSIS. In conclusion, we have developed and applied a systematic modeling workflow for pathway models that allowed us to gain insight into key mechanisms in GSIS in the pancreatic β-cell.

A simple modeling framework for prediction in the human glucose-insulin system

Forecasting blood glucose (BG) levels with routinely collected data is useful for glycemic management. BG dynamics are nonlinear, complex, and nonstationary, which can be represented by nonlinear models. However, the sparsity of routinely collected data creates parameter identifiability issues when high-fidelity complex models are used, thereby resulting in inaccurate forecasts. One can use models with reduced physiological fidelity for robust and accurate parameter estimation and forecasting with sparse data. For this purpose, we approximate the nonlinear dynamics of BG regulation by a linear stochastic differential equation: we develop a linear stochastic model, which can be specialized to different settings: type 2 diabetes mellitus (T2DM) and intensive care unit (ICU), with different choices of appropriate model functions. The model includes deterministic terms quantifying glucose removal from the bloodstream through the glycemic regulation system and representing the effect of nutrition and externally delivered insulin. The stochastic term encapsulates the BG oscillations. The model output is in the form of an expected value accompanied by a band around this value. The model parameters are estimated patient-specifically, leading to personalized models. The forecasts consist of values for BG mean and variation, quantifying possible high and low BG levels. Such predictions have potential use for glycemic management as part of control systems. We present experimental results on parameter estimation and forecasting in T2DM and ICU settings. We compare the model's predictive capability with two different nonlinear models built for T2DM and ICU contexts to have a sense of the level of prediction achieved by this model.

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.

A quasi-equilibrium reduced model of pancreatic insulin secretion

Much attention has been devoted in the last few decades to mathematical models of insulin secretion, in order to better understand the regulation of glycemia and its derangements. The glucose-insulin homeostatic mechanism is so complex and gives rise to such diverse behavior following perturbations that different models had been published, which reproduced the results of single experiments. More recently, a unifying model of pancreatic insulin secretion was proposed, which is able to account, with a single value of the (meta)parameters, for the wide array of clinically observed behavior. This model explicitly represented the pulsatile nature of the many pancreatic hormone-secreting firing units: the price to pay for its flexibility and performance is the very high dimensionality (hundreds of thousand equations) of the corresponding dynamical system. Clearly, it would be desirable to reduce this model to a much simpler form while retaining its power to reproduce heterogeneous phenomena. The present work reviews the qualitative behavior of this pancreas pulsatile model and offers some insight into its reduction in equilibrium and quasi-equilibrium conditions, also considering single-shot (non-repeated) glucose jumps from an approximately resting condition (such as would occur in standard Intra-Venous bolus dosing of glucose during diabetes diagnostic maneuvers). The resulting quasi-steady-state model can be further endowed with additional lower-order dynamics to also approximate transient behavior. Although a more accurate reduction of the original pulsatile model is left to further investigation, numerical results confirm the biomedical applicability of the formulation already obtained.

Mathematical Modeling for the Physiological and Clinical Investigation of Glucose Homeostasis and Diabetes

Mathematical modeling in the field of glucose metabolism has a longstanding tradition. The use of models is motivated by several reasons. Models have been used for calculating parameters of physiological interest from experimental data indirectly, to provide an unambiguous quantitative representation of pathophysiological mechanisms, to determine indices of clinical usefulness from simple experimental tests. With the growing societal impact of type 2 diabetes, which involves the disturbance of the glucose homeostasis system, development and use of models in this area have increased. Following the approaches of physiological and clinical investigation, the focus of the models has spanned from representations of whole body processes to those of cells, i.e., from in vivo to in vitro research. Model-based approaches for linking in vivo to in vitro research have been proposed, as well as multiscale models merging the two areas. The success and impact of models has been variable. Two kinds of models have received remarkable interest: those widely used in clinical applications, e.g., for the assessment of insulin sensitivity and β-cell function and some models representing specific aspects of the glucose homeostasis system, which have become iconic for their efficacy in describing clearly and compactly key physiological processes, such as insulin secretion from the pancreatic β cells. Models are inevitably simplified and approximate representations of a physiological system. Key to their success is an appropriate balance between adherence to reality, comprehensibility, interpretative value and practical usefulness. This has been achieved with a variety of approaches. Although many models concerning the glucose homeostasis system have been proposed, research in this area still needs to address numerous issues and tackle new opportunities. The mathematical representation of the glucose homeostasis processes is only partial, also because some mechanisms are still only partially understood. For in vitro research, mathematical models still need to develop their potential. This review illustrates the problems, approaches and contribution of mathematical modeling to the physiological and clinical investigation of glucose homeostasis and diabetes, focusing on the most relevant and stimulating models.

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 modeling informs the impact of changes in circadian rhythms and meal patterns on insulin secretion

Disruption of circadian rhythms has been associated with metabolic syndromes, including obesity and diabetes. A variety of metabolic activities are under circadian modulation, as local and global clock gene knockouts result in glucose imbalance and increased risk of metabolic diseases. Insulin release from the pancreatic β cells exhibits daily variation, and recent studies have found that insulin secretion, not production, is under circadian modulation. As consideration of daily variation in insulin secretion is necessary to accurately describe glucose-stimulated insulin secretion, we describe a mathematical model that incorporates the circadian modulation via insulin granule trafficking. We use this model to understand the effect of oscillatory characteristics on insulin secretion at different times of the day. Furthermore, we integrate the dynamics of clock genes under the influence of competing environmental signals (light/dark cycle and feeding/fasting cycle) and demonstrate how circadian disruption and meal size distribution change the insulin secretion pattern over a 24-h day.

Calcium signaling and secretory granule pool dynamics underlie biphasic insulin secretion and its amplification by glucose: experiments and modeling

Glucose-stimulated insulin secretion from pancreatic β-cells is controlled by a triggering pathway that culminates in calcium influx and regulated exocytosis of secretory granules, and by a less understood amplifying pathway that augments calcium-induced exocytosis. In response to an abrupt increase in glucose concentration, insulin secretion exhibits a first peak followed by a lower sustained second phase. This biphasic secretion pattern is disturbed in diabetes. It has been attributed to depletion and subsequent refilling of a readily releasable pool of granules or to the phasic cytosolic calcium dynamics induced by glucose. Here, we apply mathematical modeling to experimental data from mouse islets to investigate how calcium and granule pool dynamics interact to control dynamic insulin secretion. Experimental calcium traces are used as inputs in three increasingly complex models of pool dynamics, which are fitted to insulin secretory patterns obtained using a set of protocols of glucose and tolbutamide stimulation. New calcium and secretion data for so-called staircase protocols, in which the glucose concentration is progressively increased, are presented. These data can be reproduced without assuming any heterogeneity in the model, in contrast to previous modeling, because of nontrivial calcium dynamics. We find that amplification by glucose can be explained by increased mobilization and priming of granules. Overall, our results indicate that calcium dynamics contribute substantially to shaping insulin secretion kinetics, which implies that better insight into the events creating phasic calcium changes in human β-cells is needed to understand the cellular mechanisms that disturb biphasic insulin secretion in diabetes.

Estimation of insulin secretion, glucose uptake by tissues, and liver handling of glucose using a mathematical model of glucose-insulin homeostasis in lean and obese mice

Destruction of the insulin-producing β-cells is the key determinant of diabetes mellitus regardless of their types. Due to their anatomical location within the islets of Langerhans scattered throughout the pancreas, it is difficult to monitor β-cell function and mass clinically. To this end, we propose to use a mathematical model of glucose-insulin homeostasis to estimate insulin secretion, glucose uptake by tissues, and hepatic handling of glucose. We applied the mathematical model by Lombarte et al. (2013) to compare various rate constants representing glucose-insulin homeostasis between lean (11% fat)- and high fat diet (HFD; 45% fat)-fed mice. Mice fed HFD (n = 12) for 3 months showed significantly higher body weights (49.97 ± 0.52 g vs. 29.86 ± 0.46 g), fasting blood glucose levels (213.08 ± 10.35 mg/dl vs. 121.91 ± 2.26 mg/dl), and glucose intolerance compared to mice fed lean diet (n = 12). Mice were injected with 1 g/kg glucose intraperitoneally and blood glucose levels were measured at various intervals for 120 min. We performed simulation using Arena™ software based on the mathematical model and estimated the rate constants (9 parameters) for various terms in the differential equations using OptQuest™. The simulated data fit accurately to the observed data for both lean and obese mice, validating the use of the mathematical model in mice at different stages of diabetes progression. Among 9 parameters, 5 parameters including basal insulin, k₂ (rate constant for insulin-dependent glucose uptake to tissues), k₃ (rate constant for insulin-independent glucose uptake to tissues), k₄ (rate constant for liver glucose transfer), and I_pi (rate constant for insulin concentration where liver switches from glucose release to uptake) were significantly different between lean- and HFD-fed mice. Basal blood insulin levels, k₃, and I_pi were significantly elevated but k₂ and k₄ were reduced in mice fed a HFD compared to those fed a lean diet. Non-invasive assessment of the key components of glucose-insulin homeostasis including insulin secretion, glucose uptake by tissues, and hepatic handling of glucose may be helpful for individualized drug therapy and designing a customized control algorithm for the artificial pancreas.

Statistical Frailty Modeling for Quantitative Analysis of Exocytotic Events Recorded by Live Cell Imaging: Rapid Release of Insulin-Containing Granules Is Impaired in Human Diabetic β-cells

Hormones and neurotransmitters are released when secretory granules or synaptic vesicles fuse with the cell membrane, a process denoted exocytosis. Modern imaging techniques, in particular total internal reflection fluorescence (TIRF) microscopy, allow the investigator to monitor secretory granules at the plasma membrane before and when they undergo exocytosis. However, rigorous statistical approaches for temporal analysis of such exocytosis data are still lacking. We propose here that statistical methods from time-to-event (also known as survival) analysis are well suited for the problem. These methods are typically used in clinical settings when individuals are followed over time to the occurrence of an event such as death, remission or conception. We model the rate of exocytosis in response to pulses of stimuli in insulin-secreting pancreatic β-cell from healthy and diabetic human donors using piecewise-constant hazard modeling. To study heterogeneity in the granule population we exploit frailty modeling, which describe unobserved differences in the propensity to exocytosis. In particular, we insert a discrete frailty in our statistical model to account for the higher rate of exocytosis in an immediately releasable pool (IRP) of insulin-containing granules. Estimates of parameters are obtained from maximum-likelihood methods. Since granules within the same cell are correlated, i.e., the data are clustered, a modified likelihood function is used for log-likelihood ratio tests in order to perform valid inference. Our approach allows us for example to estimate the size of the IRP in the cells, and we find that the IRP is deficient in diabetic cells. This novel application of time-to-event analysis and frailty modeling should be useful also for the study of other well-defined temporal events at the cellular level.

A Novel Insulin/Glucose Model after a Mixed-Meal Test in Patients with Type 1 Diabetes on Insulin Pump Therapy

Current closed-loop insulin delivery methods stem from sophisticated models of the glucose-insulin (G/I) system, mostly based on complex studies employing glucose tracer technology. We tested the performance of a new minimal model (GLUKINSLOOP 2.0) of the G/I system to characterize the glucose and insulin dynamics during multiple mixed meal tests (MMT) of different sizes in patients with type 1 diabetes (T1D) on insulin pump therapy (continuous subcutaneous insulin infusion, CSII). The GLUKINSLOOP 2.0 identified the G/I system, provided a close fit of the G/I time-courses and showed acceptable reproducibility of the G/I system parameters in repeated studies of identical and double-sized MMTs. This model can provide a fairly good and reproducible description of the G/I system in T1D patients on CSII, and it may be applied to create a bank of “virtual” patients. Our results might be relevant at improving the architecture of upcoming closed-loop CSII systems.

A Unifying Organ Model of Pancreatic Insulin Secretion

The secretion of insulin by the pancreas has been the object of much attention over the past several decades. Insulin is known to be secreted by pancreatic β-cells in response to hyperglycemia: its blood concentrations however exhibit both high-frequency (period approx. 10 minutes) and low-frequency oscillations (period approx. 1.5 hours). Furthermore, characteristic insulin secretory response to challenge maneuvers have been described, such as frequency entrainment upon sinusoidal glycemic stimulation; substantial insulin peaks following minimal glucose administration; progressively strengthened insulin secretion response after repeated administration of the same amount of glucose; insulin and glucose characteristic curves after Intra-Venous administration of glucose boli in healthy and pre-diabetic subjects as well as in Type 2 Diabetes Mellitus. Previous modeling of β-cell physiology has been mainly directed to the intracellular chain of events giving rise to single-cell or cell-cluster hormone release oscillations, but the large size, long period and complex morphology of the diverse responses to whole-body glucose stimuli has not yet been coherently explained. Starting with the seminal work of Grodsky it was hypothesized that the population of pancreatic β-cells, possibly functionally aggregated in islets of Langerhans, could be viewed as a set of independent, similar, but not identical controllers (firing units) with distributed functional parameters. The present work shows how a single model based on a population of independent islet controllers can reproduce very closely a diverse array of actually observed experimental results, with the same set of working parameters. The model's success in reproducing a diverse array of experiments implies that, in order to understand the macroscopic behaviour of the endocrine pancreas in regulating glycemia, there is no need to hypothesize intrapancreatic pacemakers, influences between different islets of Langerhans, glycolitic-induced oscillations or β-cell sensitivity to the rate of change of glycemia.

Virtual Systems Pharmacology (ViSP) software for simulation from mechanistic systems-level models

Multiple software programs are available for designing and running large scale system-level pharmacology models used in the drug development process. Depending on the problem, scientists may be forced to use several modeling tools that could increase model development time, IT costs and so on. Therefore, it is desirable to have a single platform that allows setting up and running large-scale simulations for the models that have been developed with different modeling tools. We developed a workflow and a software platform in which a model file is compiled into a self-contained executable that is no longer dependent on the software that was used to create the model. At the same time the full model specifics is preserved by presenting all model parameters as input parameters for the executable. This platform was implemented as a model agnostic, therapeutic area agnostic and web-based application with a database back-end that can be used to configure, manage and execute large-scale simulations for multiple models by multiple users. The user interface is designed to be easily configurable to reflect the specifics of the model and the user's particular needs and the back-end database has been implemented to store and manage all aspects of the systems, such as Models, Virtual Patients, User Interface Settings, and Results. The platform can be adapted and deployed on an existing cluster or cloud computing environment. Its use was demonstrated with a metabolic disease systems pharmacology model that simulates the effects of two antidiabetic drugs, metformin and fasiglifam, in type 2 diabetes mellitus patients.

Feasibility study of a bio-inspired artificial pancreas in adults with type 1 diabetes

BACKGROUND

This study assesses proof of concept and safety of a novel bio-inspired artificial pancreas (BiAP) system in adults with type 1 diabetes during fasting, overnight, and postprandial conditions. In contrast to existing glucose controllers in artificial pancreas systems, the BiAP uses a control algorithm based on a mathematical model of β-cell physiology. The algorithm is implemented on a miniature silicon microchip within a portable hand-held device that interfaces the components of the artificial pancreas.

MATERIALS AND METHODS

In this nonrandomized open-label study each subject attended for a 6-h fasting study followed by a 13-h overnight and post-breakfast study on a separate occasion. During both study sessions the BiAP system was used, and microboluses of insulin were recommended every 5 min by the control algorithm according to subcutaneous sensor glucose levels. The primary outcome was percentage time spent in the glucose target range (3.9-10.0 mmol/L).

RESULTS

Twenty subjects (55% male; mean [SD] age, 44 [10] years; duration of diabetes, 22 [12] years; glycosylated hemoglobin, 7.4% [0.7%] [57 (7) mmol/mol]; body mass index, 25 [4] kg/m(2)) participated in the fasting study, and the median (interquartile range) percentage time in target range was 98.0% (90.8-100.0%). Seventeen of these subjects then participated in the overnight/postprandial study, where 70.7% (63.9-77.4%) of time was spent in the target range and, reassuringly, 0.0% (0.0-2.3%) of time was spent in hypoglycemia (<3.9 mmol/L).

CONCLUSIONS

The BiAP achieves safe glycemic control during fasting, overnight, and postprandial conditions.

Bio-Inspired glucose control in diabetes based on an analogue implementation of a β-cell model

This paper presents a bio-inspired method for in-vivo control of blood glucose based on a model of the pancreatic β-cell. The proposed model is shown to be implementable using low-power analogue integrated circuits in CMOS, realizing a biologically faithful implementation which captures all the behaviours seen in physiology. This is then shown to be capable of glucose control using an in silico population of diabetic subjects achieving 93% of the time in tight glycemic target (i.e., [70, 140] mg/dl) . The proposed controller is then compared with a commonly used external physiological insulin delivery (ePID) controller for glucose control. Results confirm equivalent, or superior, performance in comparison with ePID. The system has been designed in a commercially available 0.35 μm CMOS process and achieves an overall power consumption of 1.907 mW.

Minimal modeling of insulin secretion in the perfused rat pancreas: a drug effect case study

The experimental protocol of the perfused rat pancreas is commonly used to evaluate β-cell function. In this context, mathematical models become useful tools through the determination of indexes that allow the assessment of β-cell function in different experimental groups and the quantification of the effects of antidiabetic drugs, secretagogues, or treatments. However, a minimal model applicable to the isolated perfused rat pancreas has so far been unavailable. In this work, we adapt the C-peptide minimal model applied previously to the intravenous glucose tolerance test to obtain a specific model for the experimental settings of the perfused pancreas. Using the model, it is possible to estimate indexes describing β-cell responsivity for first (ΦD) and second phase (ΦS, T) of insulin secretion. The model was initially applied to untreated pancreata and afterward used for the assessment of pharmacologically relevant agents (the gut hormone GLP-1, the potent GLP-1 receptor agonist lixisenatide, and a GPR40/FFAR1 agonist, SAR1) to quantify and differentiate their effect on insulin secretion. Model fit was satisfactory, and parameters were estimated with good precision for both untreated and treated pancreata. Model application showed that lixisenatide reaches improvement of β-cell function similarly to GLP-1 (11.7- vs. 13.1-fold increase in ΦD and 2.3- vs. 2.8-fold increase in ΦS) and demonstrated that SAR1 leads to an additional improvement of β-cell function in the presence of postprandial GLP-1 levels.

Dynamics of glucose and insulin concentration connected to the β-cell cycle: model development and analysis

Background

Diabetes mellitus is a group of metabolic diseases with increased blood glucose concentration as the main symptom. This can be caused by a relative or a total lack of insulin which is produced by the β‐cells in the pancreatic islets of Langerhans. Recent experimental results indicate the relevance of the β‐cell cycle for the development of diabetes mellitus.

Methods

This paper introduces a mathematical model that connects the dynamics of glucose and insulin concentration with the β‐cell cycle. The interplay of glucose, insulin, and β‐cell cycle is described with a system of ordinary differential equations. The model and its development will be presented as well as its mathematical analysis. The latter investigates the steady states of the model and their stability.

Results

Our model shows the connection of glucose and insulin concentrations to the β‐cell cycle. In this way the important role of glucose as regulator of the cell cycle and the capability of the β‐cell mass to adapt to metabolic demands can be presented. Simulations of the model correspond to the qualitative behavior of the glucose‐insulin regulatory system showed in biological experiments.

Conclusions

This work focusses on modeling the physiological situation of the glucose‐insulin regulatory system with a detailed consideration of the β‐cell cycle. Furthermore, the presented model allows the simulation of pathological scenarios. Modification of different parameters results in simulation of either type 1 or type 2 diabetes.

Mathematical modeling of insulin secretion and the role of glucose-dependent mobilization, docking, priming and fusion of insulin granules

In this paper we develop a new mathematical model of glucose-induced insulin secretion from pancreatic islet β-cells, and we use this model to investigate the rate limiting factors. We assume that insulin granules reside in different pools inside each β-cell, and that all β-cells respond homogeneously to glucose with the same recruitment thresholds. Consistent with recent experimental observations, our model also accounts for the fusion of newcomer granules that are not pre-docked at the plasma membrane. In response to a single step increase in glucose concentration, our model reproduces the characteristic biphasic insulin release observed in multiple experimental systems, including perfused pancreata and isolated islets of rodent or human origin. From our model analysis we note that first-phase insulin secretion depends on rapid depletion of the primed, release-ready granule pools, while the second phase relies on granule mobilization from the reserve. Moreover, newcomers have the potential to contribute significantly to the second phase. When the glucose protocol consists of multiple changes in sequence (a so-called glucose staircase), our model predicts insulin spikes of increasing height, as has been seen experimentally. This increase stems from the glucose-dependent increase in the fusion rate of insulin granules at the plasma membrane of single β-cells. In contrast, previous mathematical models reproduced the staircase experiment by assuming heterogeneous β-cell activation. In light of experimental data indicating limited heterogeneous activation for β-cells within intact islets, our findings suggest that a graded, dose-dependent cell response to glucose may contribute to insulin secretion patterns observed in multiple experiments, and thus regulate in vivo insulin release. In addition, the strength of insulin granule mobilization, priming and fusion are critical limiting factors in determining the total amount of insulin release.

A bio-inspired glucose controller based on pancreatic β-cell physiology

INTRODUCTION

Control algorithms for closed-loop insulin delivery in type 1 diabetes have been mainly based on control engineering or artificial intelligence techniques. These, however, are not based on the physiology of the pancreas but seek to implement engineering solutions to biology. Developments in mathematical models of the β-cell physiology of the pancreas have described the glucose-induced insulin release from pancreatic β cells at a molecular level. This has facilitated development of a new class of bio-inspired glucose control algorithms that replicate the functionality of the biological pancreas. However, technologies for sensing glucose levels and delivering insulin use the subcutaneous route, which is nonphysiological and introduces some challenges. In this article, a novel glucose controller is presented as part of a bio-inspired artificial pancreas.

METHODS

A mathematical model of β-cell physiology was used as the core of the proposed controller. In order to deal with delays and lack of accuracy introduced by the subcutaneous route, insulin feedback and a gain scheduling strategy were employed. A United States Food and Drug Administration-accepted type 1 diabetes mellitus virtual population was used to validate the presented controller.

RESULTS

Premeal and postmeal mean ± standard deviation blood glucose levels for the adult and adolescent populations were well within the target range set for the controller [(70, 180) mg/dl], with a percent time in range of 92.8 ± 7.3% for the adults and 83.5 ± 14% for the adolescents.

CONCLUSIONS

This article shows for the first time very good glucose control in a virtual population with type 1 diabetes mellitus using a controller based on a subcellular β-cell model.

Model for glucagon secretion by pancreatic α-cells

Glucagon hormone is synthesized and released by pancreatic α-cells, one of the islet-cell types. This hormone, along with insulin, maintains blood glucose levels within the physiological range. Glucose stimulates glucagon release at low concentrations (hypoglycemia). However, the mechanisms involved in this secretion are still not completely clear. Here, using experimental calcium time series obtained in mouse pancreatic islets at low and high glucose conditions, we propose a glucagon secretion model for α-cells. Our model takes into account that the resupply of releasable granules is not only controlled by cytoplasmic , as in other neuroendocrine and endocrine cells, but also by the level of extracellular glucose. We found that, although calcium oscillations are highly variable, the average secretion rates predicted by the model fall into the range of values reported in the literature, for both stimulated and non-stimulated conditions. For low glucose levels, the model predicts that there would be a well-controlled number of releasable granules refilled slowly from a large reserve pool, probably to ensure a secretion rate that could last for several minutes. Studying the α-cell response to the addition of insulin at low glucose, we observe that the presence of insulin reduces glucagon release by decreasing the islet level. This observation is in line with previous work reporting that dynamics, mainly frequency, is altered by insulin [1]. Thus, the present results emphasize the main role played by and glucose in the control of glucagon secretion by α-cells. Our modeling approach also shows that calcium oscillations potentiate glucagon secretion as compared to constant levels of this cellular messenger. Altogether, the model sheds new light on the subcellular mechanisms involved in α-cell exocytosis, and provides a quantitative predictive tool for studying glucagon secretion modulators in physiological and pathological conditions.

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.

Coupling of metabolic, second messenger pathways and insulin granule dynamics in pancreatic beta-cells: a computational analysis

Insulin secretory responses to nutrient stimuli and hormonal modulators in pancreatic beta-cells are controlled by a variety of secondary messengers. We have analyzed numerous mechanisms responsible for regulated exocytosis in these cells and present an integrated mathematical model of cytosolic Ca²⁺, cAMP and granule dynamics. The insulin-containing granules in the beta-cell were divided into four classes: a large "reserve" granule pool, a smaller pool of the morphologically docked granules that is chemically 'primed' for release or the "readily releasable pool", and a pool of "restless newcomer granules" that undergoes preferential exocytosis. The model incorporates glucose and other aspects of metabolism, the cAMP amplifying pathway, insulin granule dynamics and the exocyst concept for granule binding. The values of most of the model parameters were inferred from available experimental data. The model can generate both the fast first phase and slow biphasic insulin secretion found experimentally in response to a step increase of membrane potential or of glucose. The numerical simulations have also reproduced a variety of experimental conditions, such as periodic stimulation by high K⁺ and the potentiation induced in islets by pre-incubation with cAMP pathway activators. The explicit incorporation of Ca²⁺ channels, Ca²⁺ and cAMP dynamics allows the model to be further connected to current models for calcium and metabolic dynamics and provides an interpretation of the roles of the triggering and amplifying pathways of glucose-stimulated insulin secretion. The model may be important in the identification of pharmacological targets for improving insulin secretion in type 2 diabetes.

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

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

Modeling mechanisms of cell secretion

Secretion is a fundamental cellular process involving the regulated release of intracellular products from cells. Physiological functions such as neurotransmission, or the release of hormones and digestive enzymes, are all governed by cell secretion. Anomalies in the processes involved in secretion contribute to the development and progression of diseases such as diabetes and other hormonal disorders. To unravel the mechanisms that govern such diseases, it is essential to understand how hormones, growth factors and neurotransmitters are synthesized and processed, and how their signals are recognized, amplified and transmitted by intracellular signaling pathways in the target cells. Here, we discuss diverse aspects of the detailed mechanisms involved in secretion based on mathematical models. The models range from stochastic ones describing the trafficking of secretory vesicles to deterministic ones investigating the regulation of cellular processes that underlie hormonal secretion. In all cases, the models are closely related to experimental results and suggest theoretical predictions for the secretion mechanisms.

The Unscented Kalman Filter estimates the plasma insulin from glucose measurement

Understanding the simultaneous interaction within the glucose and insulin homeostasis in real-time is very important for clinical treatment as well as for research issues. Until now only plasma glucose concentrations can be measured in real-time. To support a secure, effective and rapid treatment e.g. of diabetes a real-time estimation of plasma insulin would be of great value. A novel approach using an Unscented Kalman Filter that provides an estimate of the current plasma insulin concentration is presented, which operates on the measurement of the plasma glucose and Bergman's Minimal Model of the glucose insulin homeostasis. We can prove that process observability is obtained in this case. Hence, a successful estimator design is possible. Since the process is nonlinear we have to consider estimates that are not normally distributed. The symmetric Unscented Kalman Filter (UKF) will perform best compared to other estimator approaches as the Extended Kalman Filter (EKF), the simplex Unscented Kalman Filter (UKF), and the Particle Filter (PF). The symmetric UKF algorithm is applied to the plasma insulin estimation. It shows better results compared to the direct (open loop) estimation that uses a model of the insulin subsystem.

Metabolic amplifying pathway increases both phases of insulin secretion independently of β-cell actin microfilaments

Two pathways control glucose-induced insulin secretion (IS) by β-cells. The triggering pathway involves ATP-sensitive potassium (KATP) channel-dependent depolarization, Ca2+ influx, and a rise in the cytosolic Ca2+ concentration ([Ca2+]c), which triggers exocytosis of insulin granules. The metabolic amplifying pathway augments IS without further increasing [Ca2+]c. The underlying mechanisms are unknown. Here, we tested the hypothesis that amplification implicates actin microfilaments. Mouse islets were treated with latrunculin B and cytochalasin B to depolymerize actin or jasplakinolide to polymerize actin. They were then perifused to measure [Ca2+]c and IS. Metabolic amplification was studied during imposed steady elevation of [Ca2+]c by tolbutamide or KCl or by comparing the magnitude of [Ca2+]c and IS changes produced by glucose and tolbutamide. Both actin polymerization and depolymerization augmented IS triggered by all stimuli without increasing (sometimes decreasing) [Ca2+]c, which indicates a predominantly inhibitory function of microfilaments in exocytosis at a step distal to [Ca2+]c increase. When [Ca2+]c was elevated and controlled by KCl or tolbutamide, the amplifying action of glucose was facilitated by actin depolymerization and unaffected by polymerization. Both phases of IS were larger in response to high-glucose than to tolbutamide in low-glucose, although triggering [Ca2+]c was lower. This difference in IS, due to amplification, persisted when the IS rate was doubled by actin depolymerization or polymerization. In conclusion, metabolic amplification is rapid and influences the first as well as the second phase of IS. It is a late step of stimulus-secretion coupling, which does not require functional actin microfilaments and could correspond to acceleration of the priming process conferring release competence to insulin granules.

Lessons from models of pancreatic beta cells for engineering glucose-sensing cells

Mathematical models of pancreatic beta cells suggest design principles that can be applied to engineering cells to sense glucose and secrete insulin. Engineering cells can potentially both contribute to future diabetes therapies and generate new insights into beta-cell function. The focus is on ion channels, Ca(2+)handling, and elements of metabolism that combine to produce the varied oscillatory patterns exhibited by beta cells.

Cellular modeling: insight into oral minimal models of insulin secretion

The oral glucose tolerance test and meal tolerance test are common clinical tests of the glucose-insulin system. Several mathematical models have been suggested as means to extract information about beta-cell function from data from oral tolerance tests. Any such model needs to be fairly simple but should at the same time be linked to the underlying biology of the insulin-secreting beta-cells. The scope of the present work is to present a way to make such a connection using a recent model describing intracellular mechanisms. We show how the three main components of oral minimal secretion models, derivative control, proportional control, and delay, are related to subcellular events, thus providing mechanistic underpinning of the assumptions of the minimal models.

Regulation of insulin secretion: a matter of phase control and amplitude modulation

The consensus model of stimulus-secretion coupling in beta cells attributes glucose-induced insulin secretion to a sequence of events involving acceleration of metabolism, closure of ATP-sensitive K(+) channels, depolarisation, influx of Ca(2+) and a rise in cytosolic free Ca(2+) concentration ([Ca(2+)](c)). This triggering pathway is essential, but would not be very efficient if glucose did not also activate a metabolic amplifying pathway that does not raise [Ca(2+)](c) further but augments the action of triggering Ca(2+) on exocytosis. This review discusses how both pathways interact to achieve temporal control and amplitude modulation of biphasic insulin secretion. First-phase insulin secretion is triggered by the rise in [Ca(2+)](c) that occurs synchronously in all beta cells of every islet in response to a sudden increase in the glucose concentration. Its time course and duration are shaped by those of the Ca(2+) signal, and its amplitude is modulated by the magnitude of the [Ca(2+)](c) rise and, substantially, by amplifying mechanisms. During the second phase, synchronous [Ca(2+)](c) oscillations in all beta cells of an individual islet induce pulsatile insulin secretion, but these features of the signal and response are dampened in groups of intrinsically asynchronous islets. Glucose has hardly any influence on the amplitude of [Ca(2+)](c) oscillations and mainly controls the time course of triggering signal. Amplitude modulation of insulin secretion pulses largely depends on the amplifying pathway. There are more similarities than differences between the two phases of glucose-induced insulin secretion. Both are subject to the same dual, hierarchical control over time and amplitude by triggering and amplifying pathways, suggesting that the second phase is a sequence of iterations of the first phase.

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

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

A subcellular model of glucose-stimulated pancreatic insulin secretion

When glucose is raised from a basal to stimulating level, the pancreatic islets respond with a typical biphasic insulin secretion pattern. Moreover, the pancreas is able to recognize the rate of change of the glucose concentration. We present a relatively simple model of insulin secretion from pancreatic beta-cells, yet founded on solid physiological grounds and capable of reproducing a series of secretion patterns from perfused pancreases as well as from stimulated islets. The model includes the notion of distinct pools of granules as well as mechanisms such as mobilization, priming, exocytosis and kiss-and-run. Based on experimental data, we suggest that the individual beta-cells activate at different glucose concentrations. The model reproduces most of the data it was tested against very well, and can therefore serve as a general model of glucose-stimulated insulin secretion. Simulations predict that the effect of an increased frequency of kiss-and-run exocytotic events is a reduction in insulin secretion without modification of the qualitative pattern. Our model also appears to be the first physiology-based one to reproduce the staircase experiment, which underlies 'derivative control', i.e. the pancreatic capacity of measuring the rate of change of the glucose concentration.

Identifying the targets of the amplifying pathway for insulin secretion in pancreatic beta-cells by kinetic modeling of granule exocytosis

A kinetic model for insulin secretion in pancreatic beta-cells is adapted from a model for fast exocytosis in chromaffin cells. The fusion of primed granules with the plasma membrane is assumed to occur only in the "microdomain" near voltage-sensitive L-type Ca(2+)-channels, where [Ca(2+)] can reach micromolar levels. In contrast, resupply and priming of granules are assumed to depend on the cytosolic [Ca(2+)]. Adding a two-compartment model to handle the temporal distribution of Ca(2+) between the microdomain and the cytosol, we obtain a unified model that can generate both the fast granule fusion and the slow insulin secretion found experimentally in response to a step of membrane potential. The model can simulate the potentiation induced in islets by preincubation with glucose and the reduction in second-phase insulin secretion induced by blocking R-type Ca(2+)-channels (Ca(V)2.3). The model indicates that increased second-phase insulin secretion induced by the amplifying signal is controlled by the "resupply" step of the exocytosis cascade. In contrast, enhancement of priming is a good candidate for amplification of first-phase secretion by glucose, cyclic adenosine 3':5'-cyclic monophosphate, and protein kinase C. Finally, insulin secretion is enhanced when the amplifying signal oscillates in phase with the triggering Ca(2+)-signal.