Michela Riz

Mathematical Modeling of Heterogeneous Electrophysiological Responses in Human β-Cells

Electrical activity plays a pivotal role in glucose-stimulated insulin secretion from pancreatic -cells. Recent findings have shown that the electrophysiological characteristics of human -cells differ from their rodent counterparts. We show that the electrophysiological responses in human -cells to a range of ion channels antagonists are heterogeneous. In some cells, inhibition of small-conductance potassium currents has no effect on action potential firing, while it increases the firing frequency dramatically in other cells. Sodium channel block can sometimes reduce action potential amplitude, sometimes abolish electrical activity, and in some cells even change spiking electrical activity to rapid bursting. We show that, in contrast to L-type -channels, P/Q-type -currents are not necessary for action potential generation, and, surprisingly, a P/Q-type -channel antagonist even accelerates action potential firing. By including SK-channels and dynamics in a previous mathematical model of electrical activity in human -cells, we investigate the heterogeneous and nonintuitive electrophysiological responses to ion channel antagonists, and use our findings to obtain insight in previously published insulin secretion measurements. Using our model we also study paracrine signals, and simulate slow oscillations by adding a glycolytic oscillatory component to the electrophysiological model. The heterogenous electrophysiological responses in human -cells must be taken into account for a deeper understanding of the mechanisms underlying insulin secretion in health and disease, and as shown here, the interdisciplinary combination of experiments and modeling increases our understanding of human -cell physiology.

Non-invasive continuous glucose monitoring: improved accuracy of point and trend estimates of the Multisensor system

Non-invasive continuous glucose monitoring (NI-CGM) sensors are still at an early stage of development, but, in the near future, they could become particularly appealing in diabetes management. Solianis Monitoring AG (Zurich, Switzerland) has proposed an approach for NI-CGM based on a multi-sensor concept, embedding primarily dielectric spectroscopy and optical sensors. This concept requires a mathematical model able to estimate glucose levels from the 150 channels directly measured through the Multisensor. A static multivariate linear regression model (with order and parameters common to the entire population of subjects) was proposed for such a scope (Caduff et al., Biosens Bioelectron 26:3794–3800, 2011). The aim of this work is to evaluate the accuracy in the estimation of glucose levels and trends that the NI-CGM Multisensor platform can achieve by exploiting different techniques for model identification, namely, ordinary least squares, subset variable selection, partial least squares and least absolute shrinkage and selection operator (LASSO). Data collected in human beings monitored for a total of 45 study days were used for model identification and model test. Several metrics of standard use in the diabetes scientific community to measure point and clinical accuracy of glucose sensors were used to assess the models. Results indicate that the LASSO technique is superior to the others shrinking many channel weights to zero thus leading to smoother glucose profiles and resulting in a more robust model to possible artifacts in the Multisensor data. Although, as expected, the performance of the NI-CGM system with the LASSO model is not yet comparable with that of enzyme-based needle glucose sensors, glucose trends are satisfactorily estimated. Considering the non-invasive nature of the multi-sensor platform, this result can have an immediate impact in the current clinical practice, e.g., to integrate sparse self-monitoring of blood glucose data with an indication of the glucose trend to aid the diabetic patient in dealing with, or even preventing in the short time scale, the threats of critical events such as hypoglycaemia.

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

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

Mathematical Modeling of Interacting Glucose-Sensing Mechanisms and Electrical Activity Underlying Glucagon-Like Peptide 1 Secretion

Intestinal L-cells sense glucose and other nutrients, and in response release glucagon-like peptide 1 (GLP-1), peptide YY and other hormones with anti-diabetic and weight-reducing effects. The stimulus-secretion pathway in L-cells is still poorly understood, although it is known that GLP-1 secreting cells use sodium-glucose co-transporters (SGLT) and ATP-sensitive K+-channels (K(ATP)-channels) to sense intestinal glucose levels. Electrical activity then transduces glucose sensing to Ca2+-stimulated exocytosis. This particular glucose-sensing arrangement with glucose triggering both a depolarizing SGLT current as well as leading to closure of the hyperpolarizing K(ATP) current is of more general interest for our understanding of glucose-sensing cells. To dissect the interactions of these two glucose-sensing mechanisms, we build a mathematical model of electrical activity underlying GLP-1 secretion. Two sets of model parameters are presented: one set represents primary mouse colonic L-cells; the other set is based on data from the GLP-1 secreting GLUTag cell line. The model is then used to obtain insight into the differences in glucose-sensing between primary L-cells and GLUTag cells. Our results illuminate how the two glucose-sensing mechanisms interact, and suggest that the depolarizing effect of SGLT currents is modulated by K(ATP)-channel activity. Based on our simulations, we propose that primary L-cells encode the glucose signal as changes in action potential amplitude, whereas GLUTag cells rely mainly on frequency modulation. The model should be useful for further basic, pharmacological and theoretical investigations of the cellular signals underlying endogenous GLP-1 and peptide YY release.

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.

Assessment of linear regression techniques for modeling multisensor data for non-invasive continuous glucose monitoring

New scenarios in diabetes treatment have been opened in the last ten years by continuous glucose monitoring (CGM) sensors. In particular, Non-Invasive CGM sensors are particularly appealing, even though they are still at an early stage of development. Solianis Monitoring AG (Zürich, Switzerland) has proposed an approach based on a multisensor concept, embedding primarily dielectric spectroscopy and optical sensors. This concept requires a mathematical model able to reconstruct the glucose concentration from the 150 channels measured with the device. Assuming a multivariate linear regression model (valid and usable for different individuals), the aim of this paper is the assessment of some techniques usable for determining such a model, namely Ordinary Least Squares (OLS), Partial Least Squares (PLS) and Least Absolute Shrinkage and Selection Operator (LASSO). Once the model is identified on a training set, the accuracy of prospective glucose profiles estimated from ”unseen” multisensor data is assessed. Preliminary results obtained from 18 in-clinic study days show that sufficiently accurate reconstruction of glucose levels can be achieved if suitable model identification techniques, such as LASSO, are considered.

Inwardly rectifying Kir2.1 currents in human β-cells control electrical activity: characterisation and mathematical modelling.

Pancreatic β-cells fire action potentials as do cardiac cells and neurons, and electrical activity plays a central role in glucose-stimulated insulin secretion, which is disturbed in diabetes. The inwardly rectifying Kir2.1 potassium channels (KCNJ2 gene) control cardiac electrical activity by stabilising the interspike interval. Loss-of-function abnormalities in cardiac Kir2.1 currents can lead to the long QT syndrome and alterations of cardiac excitability, and patients with some forms of long QT syndrome suffer from over-secretion of insulin, hyperinsulinemia and symptomatic hypoglycemia. The KCNJ2 gene is also expressed in human pancreatic islets, and we show that functional Kir2.1 currents are present in human β-cells. We characterised the human Kir2.1 β-cell current, and included it in a recent mathematical model of electrical activity in human β-cells. Based on our simulations we propose that Kir2.1 currents control the interspike interval, and predict that blocking Kir2.1 channels increases the action potential frequency, which should augment the rate of insulin secretion. Vice versa, the model suggests that hyperactive Kir2.1 channels may lead to reduced insulin secretion. Our findings provide a putative link between increased insulin secretion and the long QT syndrome, and give novel insight into normal and disturbed β-cell function.

An index of parameter reproducibility accounting for estimation uncertainty: theory and case study on β-cell responsivity and insulin sensitivity.

Parameter reproducibility is necessary to perform longitudinal studies where parameters are assessed to monitor disease progression or effect of therapy but are also useful in powering the study, i.e., to define how many subjects should be studied to observe a given effect. The assessment of parameter reproducibility is usually accomplished by methods that do not take into account the fact that these parameters are estimated with uncertainty. This is particularly relevant in physiological and clinical studies where usually reproducibility cannot be assessed by multiple testing and is usually assessed from a single replication of the test. Working in a suitable stochastic framework, here we propose a new index (S) to measure reproducibility that takes into account parameter uncertainty and is particularly suited to handle the normal testing conditions of physiological and clinical investigations. Simulation results prove that S, by properly taking into account parameter uncertainty, is more accurate and robust than the methods available in the literature. The new metric is applied to assess reproducibility of insulin sensitivity and β-cell responsivity of a mixed-meal tolerance test from data obtained in the same subjects retested 1 wk apart. Results show that the indices of insulin sensitivity and β-cell responsivity to glucose are well reproducible. We conclude that the oral minimal models provide useful indices that can be used safely in prospective studies or to assess the efficacy of a given therapy.

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

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