Insulin release at the molecular level: metabolic-electrophysiological modeling of the pancreatic beta-cells

A primer on modelling pancreatic islets: from models of coupled β-cells to multicellular islet models

Pancreatic islets are mini-organs composed of hundreds or thousands of ɑ, β and δ-cells, which, respectively, secrete glucagon, insulin and somatostatin, key hormones for the regulation of blood glucose. In pancreatic islets, hormone secretion is tightly regulated by both internal and external mechanisms, including electrical communication and paracrine signaling between islet cells. Given its complexity, the experimental study of pancreatic islets has been complemented with computational modeling as a tool to gain a better understanding about how all the mechanisms involved at different levels of organization interact. In this review, we describe how multicellular models of pancreatic cells have evolved from the early models of electrically coupled β-cells to models in which experimentally derived architectures and both electrical and paracrine signals have been considered.

Identification of structures for ion channel kinetic models

Markov models of ion channel dynamics have evolved as experimental advances have improved our understanding of channel function. Past studies have examined limited sets of various topologies for Markov models of channel dynamics. We present a systematic method for identification of all possible Markov model topologies using experimental data for two types of native voltage-gated ion channel currents: mouse atrial sodium currents and human left ventricular fast transient outward potassium currents. Successful models identified with this approach have certain characteristics in common, suggesting that aspects of the model topology are determined by the experimental data. Incorporating these channel models into cell and tissue simulations to assess model performance within protocols that were not used for training provided validation and further narrowing of the number of acceptable models. The success of this approach suggests a channel model creation pipeline may be feasible where the structure of the model is not specified a priori.

Markov models of ion channel dynamics have evolved as experimental advances have improved our understanding of channel function. Past studies have examined limited sets of various structures for Markov models of channel dynamics. Here, we present a computational routine designed to thoroughly search for Markov model topologies for simulating whole-cell currents. We tested this method on two distinct types of voltage-gated cardiac ion channels and found the number of states and connectivity required to recapitulate experimentally observed kinetics. Successful models identified with this approach have certain characteristics in common, suggesting that model structures are determined by the experimental data. Incorporation of these models into higher scale action potential and cable (an approximation of one-dimensional action potential propagation) simulations, identified key channel phenomena that were required for proper function. These methods provide a route to create functional channel models that can be used for action potential simulation without pre-defining their structure ahead of time.

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.

Phase transitions in pancreatic islet cellular networks and implications for type-1 diabetes

In many aspects the onset of a chronic disease resembles a phase transition in a complex dynamic system: Quantitative changes accumulate largely unnoticed until a critical threshold is reached, which causes abrupt qualitative changes of the system. In this study we examine a special case, the onset of type-1 diabetes (T1D), a disease that results from loss of the insulin-producing pancreatic islet β cells. Within each islet, the β cells are electrically coupled to each other via gap-junctional channels. This intercellular coupling enables the β cells to synchronize their insulin release, thereby generating the multiscale temporal rhythms in blood insulin that are critical to maintaining blood glucose homeostasis. Using percolation theory we show how normal islet function is intrinsically linked to network connectivity. In particular, the critical amount of β-cell death at which the islet cellular network loses site percolation is consistent with laboratory and clinical observations of the threshold loss of β cells that causes islet functional failure. In addition, numerical simulations confirm that the islet cellular network needs to be percolated for β cells to synchronize. Furthermore, the interplay between site percolation and bond strength predicts the existence of a transient phase of islet functional recovery after onset of T1D and introduction of treatment, potentially explaining the honeymoon phenomenon. Based on these results, we hypothesize that the onset of T1D may be the result of a phase transition of the islet β-cell network.

A practical quantification of blood glucose production due to high-level chronic stress

Blood glucose (BG) is the primary metabolic fuel for, among others, cancer cell progression, cardiovascular disease and inflammation. Stress is an important contributor to the amount of BG produced especially by the liver. In this paper, we attempt to quantify the BG production due to chronic (in the order of weeks) high-level psychological stress in a manner that a lay person will understand. Three independent approaches were used. The first approach was based on a literature survey of stress hormone data from healthy individuals and its subsequent mathematical manipulation. The next approach was a deductive process where BG levels could be deduced from published stress data of large cardiovascular clinical trials. The third approach used empirical BG data and a BG simulation model. The three different methods produced an average BG increase of 2.2-fold above basal for high levels of stress over a period of more than a day. The standard deviation normalized to the average value was 4.5%.

A mathematical model of β-cells in an islet of Langerhans sensing a glucose gradient

Pancreatic β-cells release insulin in response to increased glucose levels. Compared to isolated β-cells, β-cells embedded within the islets of Langerhans network exhibit a coordinated and greater insulin secretion response to glucose. This coordinated activity is considered to rely on gap-junctions. We investigated the β-cell electrophysiology and the calcium dynamics in islets in response to glucose gradients. While at constant glucose the network of β-cells fires in a correlated fashion, a glucose gradient induces a sharp division into an active and an inactive part. We hypothesized that this sharp transition is mediated by the specific properties of the gap-junctions. We used a mathematical model of the β-cell electrophysiology in islets to discuss possible origins of this sharp transition in electrical activity. In silico, gap-junctions were required for such a transition. However, the small width of transition was only found when a stochastic variability of the expression of key transmembrane proteins, such as the ATP-dependent potassium channel, was included. The agreement with experimental data was further improved by assuming a delay of gap-junction currents, which points to a role of spatial constraints in the β-cell. This result clearly demonstrates the power of mathematical modeling in disentangling causal relationships in complex systems.

LIBS-based detection of antioxidant elements: a new strategy

The present study deals with the scientific evaluation of antioxidant potential of aqueous extract of Trichosanthes dioica fruits on diabetes-induced oxidative stress of diabetic rats. The most effective dose of mg/kg bw of fruit aqueous extract was given orally to diabetic rats for 30 days. Different oxidative stress parameters were analyzed in various tissues of control and treated diabetic rats. The observed elevated level of lipid peroxidation (LPO) comes down significantly (p < 0.05) and decreased activities of antioxidant enzymes such as catalase (CAT), superoxide dismutase (SOD), glutathione peroxidase (GPx), and glutathione-S-transferase (GST) got increased (p < 0.05) significantly of diabetic rats on extract treatment. Laser-Induced Breakdown Spectroscopy (LIBS) has been used as an analytical tool to detect major and minor elements like Mg, Fe, Na, K, Zn, Ca, H, O, C, and N present in the extract. The higher concentration of Ca(2+), Mg(2+), and Fe(2+), as reflected by their intensities are responsible for antioxidant potential of T. dioica.

A new model to estimate bolus insulin need

BACKGROUND

Patients with type 1 diabetes usually use the carbohydrate (CHO) counting method to establish their bolus insulin need. However, most still struggle with blood glucose control. We believe that for good control their insulin requirements should strive to mimic the insulin secretion of those without diabetes. The objective here is to develop, from first principles, a better model than the CHO counting model for calculating the bolus insulin need of subjects without diabetes. Such a model may also provide better blood glucose control for patients with type 1 diabetes. This will be investigated in a future article.

METHODS

Equations for metabolism of CHO and for insulin response were derived from first principles. Clinical trials were used to verify these models. The final results--namely, the new bolus insulin requirement equations--were verified using clinical trials by other researchers (Wolever and co-workers). Their methods used are described in their articles given in the list of references.

RESULTS

The postprandial insulin secretion relationships resulted in an average Pearson R(2) of 0.807 (for the new method) versus the old method's R(2) of 0.562 (CHO counting).

CONCLUSIONS

The newly derived equation provides a better approximation than the CHO counting method of insulin secretion due to metabolized blood glucose energy from ingested carbohydrates for those without diabetes. We believe that insulin dosage requirements for a patient with type 1 diabetes should mimic the insulin secretion of those without diabetes. If this is true, it means that the new equation should also estimate bolus insulin need for a patient with type 1 diabetes more accurately than before. This will be investigated in a future article.

Modeling of oscillatory bursting activity of pancreatic beta-cells under regulated glucose stimulation

A mathematical model to describe the oscillatory bursting activity of pancreatic beta-cells is combined with a model of glucose regulation system in this work to study the bursting pattern under regulated extracellular glucose stimulation. The bursting electrical activity in beta-cells is crucial for the release of insulin, which acts to regulate the blood glucose level. Different types of bursting pattern have been observed experimentally in glucose-stimulated islets both in vivo and in vitro, and the variations in these patterns have been linked to changes in glucose level. The combined model in this study enables us to have a deeper understanding on the regime change of bursting pattern when glucose level changes due to hormonal regulation, especially in the postprandial state. This is especially important as the oscillatory components of electrical activity play significant physiological roles in insulin secretion and some components have been found to be lost in type 2 diabetic patients.

Antidiabetic effect of Withania coagulans in experimental rats

The present study defines the systematic evaluation and the role of minerals in glycemic potential of aqueous extract of Withania coagulans fruits in order to develop an effective and safe alternative treatment for diabetes mellitus. Laser Induced Breakdown Spectroscopy was used for glycemic element detection. The study is based on the results of lowering in blood glucose levels of normal, sub, mild and severely diabetic rats assessed during fasting blood glucose, glucose tolerance test and post prandial glucose studies. The dose of 1000mg/ kg was identified as the most effective dose, which reduces the Fasting Blood Glucose level maximum by 33.2% at 4h in normal rats during fasting blood glucose studies. Glucose tolerance test studies of normal, sub and mild diabetic rats showed the maximum reduction of 15.7, 28.9 and 37.8% at 3h respectively. Long-term study in case of severely diabetic rats showed reduction of 52.9 and 54.1% in Fasting Blood Glucose and Post Prandial Glucose levels respectively after 30 days of treatment. The present study, besides confirming hypoglycemic and antidiabetic activities of aqueous extract of W. coagulans, helps in identifying the role of trace minerals like Mg & Ca responsible for antidiabetic potential of this potent indigenous shrub.

Antidiabetic effect of Ficus bengalensis aerial roots in experimental animals

ETHNOPHARMACOLOGICAL RELEVANCE

Herbal preparations of Ficus bengalensis had been considered as effective, economical and safe ethnomedicines for various ailments in Indian traditional system of medicine.

AIM OF STUDY

The present study was aimed to explore scientifically the antidiabetic potential of Ficus bengalensis aerial roots as its bark had already been reported to possess antidiabetic efficacy.

MATERIALS AND METHODS

Effect of variable doses of aqueous extract of Ficus bengalensis aerial roots on blood glucose level (BGL) of normal-, sub- and mild-diabetic models have been studied and the results were compared with the reference drug Glipizide and elemental Mg and Ca intake as glycemic elements.

RESULTS

The dose of 300 mg kg(-1) showed the maximum fall of 43.8 and 40.7% in BGL during FBG and glucose tolerance test (GTT) studies of normal rats, respectively. The same dose showed a marked reduction in BGL of 54.3% in sub- and 51.7% in mild-diabetic rats during GTT. The concentration of Mg (1.02%) and Ca (0.85%) identified through laser induced breakdown spectroscopy (LIBS) in the most effective dose could be responsible for this high percentage fall in BGL as they take part in glucose metabolism.

CONCLUSION

The hypoglycemic effect in normoglycemic and antidiabetic effect in sub- and mild-diabetic models of aqueous extract of aerial roots of Ficus bengalensis are due to the presence of these glycemic elements in high concentration with respect to other elements.

A new biphasic minimal model

Plasma insulin levels vary in an oscillatory fashion in both basal and postprandial states. The basic pattern is one of rapid (10 min) pulses superimposed on slower (50-100 min) oscillations. These oscillations increase after a glucose load and are altered in type 2 diabetes, impaired glucose tolerance or ageing. In response to a square-wave increase in interstitial glucose, beta-cells release insulin in a biphasic manner, with a sharp first phase lasting approximately 10 min followed by a gradually increasing release (second phase). Both phases are important for maintaining glucose homeostasis, but more emphasis has been placed on early insulin release because its attenuation causes glucose intolerance and late hyperinsulinemia. A new minimal model of glucose and insulin concentrations in plasma, which incorporates both the pulsatile and biphasic aspect of insulin production into existing minimal models, has been developed. The model is founded upon recent results on the action of beta-cells as fuel sensors and the dynamics of secretory granule exocytosis. The inclusion of a flexible model of insulin release is essential if the model is to be used to describe diabetic patients for more than a few hours and is a step towards a 24 hour free living model.

Modeling of Ca2+ flux in pancreatic beta-cells: role of the plasma membrane and intracellular stores

We have developed a detailed mathematical model of ionic flux in beta-cells that includes the most essential channels and pumps in the plasma membrane. This model is coupled to equations describing Ca2+, inositol 1,4,5-trisphosphate (IP3), ATP, and Na+ homeostasis, including the uptake and release of Ca2+ by the endoplasmic reticulum (ER). In our model, metabolically derived ATP activates inward Ca2+ flux by regulation of ATP-sensitive K+ channels and depolarization of the plasma membrane. Results from the simulations support the hypothesis that intracellular Na+ and Ca2+ in the ER can be the main variables driving both fast (2-7 osc/min) and slow intracellular Ca2+ concentration oscillations (0.3-0.9 osc/min) and that the effect of IP3 on Ca2+ leak from the ER contributes to the pattern of slow calcium oscillations. Simulations also show that filling the ER Ca2+ stores leads to faster electrical bursting and Ca2+ oscillations. Specific Ca2+ oscillations in isolated beta-cell lines can also be simulated.

Storage capacity of attractor neural networks with depressing synapses

We compute the capacity of a binary neural network with dynamic depressing synapses to store and retrieve an infinite number of patterns. We use a biologically motivated model of synaptic depression and a standard mean-field approach. We find that at T=0 the critical storage capacity decreases with the degree of the depression. We confirm the validity of our main mean-field results with numerical simulations.

Mathematical modeling in glucose metabolism and insulin secretion

PURPOSE OF REVIEW

Mathematical models in the study of glucose metabolism, insulin secretion and the insulin-glucose interactions have a longstanding tradition. The recent advances in this area are reviewed, with particular emphasis on the methods for the assessment of insulin sensitivity and insulin secretion. The available models are illustrated, and their common aspects and differences discussed.

RECENT FINDINGS

For the assessment of insulin sensitivity and beta-cell function, several modeling methods have recently been developed. Models for insulin sensitivity provide insulin-sensitivity indices from simple clinical tests, or a rich multiple-parameter characterization of insulin sensitivity from more elaborate experiments. Models for beta-cell function yield indices that quantify the ability of the beta-cells to respond to glucose stimuli. Furthermore, models of the insulin-glucose interactions propose interesting explanations of some experimental observations such as insulin-glucose oscillations and the progression to type 2 diabetes.

SUMMARY

Mathematical models in this area continue to evolve toward more accurate and clinically applicable approaches, and should be considered as a useful resource for clinical investigators. Models also have a potentially important role for understanding the mechanisms governing the insulin-glucose regulation system.