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

Pancreatic islets are mini-organs composed of hundreds or thousands of ɑ, ß 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 ß-cells to models in which experimentally derived architectures and both electrical and paracrine signals have been considered.

Computational Reconstruction of Pancreatic Islets as a Tool for Structural and Functional Analysis.

Structural properties of pancreatic islets are key for the functional response of insulin, glucagon, and somatostatin-secreting cells, due to their implications in intraislet communication via electric, paracrine, and autocrine signaling. In this protocol, the three-dimensional architecture of a pancreatic islet is firstly reconstructed from experimental data using a novel computational algorithm. Next, the morphological and connectivity properties of the reconstructed islet, such as the number and percentages of the different type of cells, cellular volume, and cell-to-cell contacts, are obtained. Then, network theory is used to describe the connectivity properties of the islet through network-derived metrics such as average degree, clustering coefficient, density, diameter, and efficiency. Finally, all these properties are functionally evaluated through computational simulations using a model of coupled oscillators. Overall, here we describe a step-by-step workflow, implemented in IsletLab, a multiplatform application developed specifically for the study and simulation of pancreatic islets, to apply a novel computational methodology to characterize and analyze pancreatic islets as a complement to the experimental work.

Deterministic modeling of single-channel and whole-cell currents.

As a complement to the experimental work, mathematical models are extensively used to study the functional properties of ionic channels. Even though it is generally assumed that the gating of ionic channels is a Markovian phenomenon, reports based on non-traditional analyses of experimental recordings suggest that non-Markovian processes might be also present. While the stochastic Markov models are by far the most adopted approach for the modeling of ionic channels, a model based on the idea of a deterministic process underlying the gating of ionic channels was proposed by Liebovitch and Toth (Liebovitch, L.S. and Toth, T.I., 1991. Journal of Theoretical Biology, 148(2), pp.243-267.) Here, by using a voltage-dependent K+ channel as a first approximation, we propose a modified version of the deterministic model of Liebovitch and Toth that, in addition to reproducing the single-channel currents simulated by a two-states Markov model, it is capable of reproducing the whole-cell currents produced by a population of K+ channels.

Reconstructing human pancreatic islet architectures using computational optimization.

We outline a general methodology based on computational optimization and experimental data to reconstruct human pancreatic islet architectures. By using the nuclei coordinates of islet cells obtained through DAPI staining, cell types identified by immunostaining, and cell size distributions estimated from capacitance measurements, reconstructed islets composed of non-overlapping spherical cells were obtained through an iterative optimization procedure. In all cases, the reconstructed architectures included >99% of the experimental identified cells, each of them having a radius within the experimentally reported ranges. Given the wide use of mathematical modeling for the study of pancreatic cells, and recently, of cell-cell interactions within the pancreatic islets, the methodology here proposed, also capable of identifying cell-to-cell contacts, is aimed to provide with a framework for modeling and analyzing experimentally-based pancreatic islet architectures.

Electrophysiological models of the human pancreatic δ-cell: From single channels to the firing of action potentials.

Minimal mathematical models were developed to describe the electrophysiological properties of human δ-cells. Markov models of single channels were first developed based on the analysis of electrophysiological data. Monte Carlo simulations of voltage-clamp experiments were performed in an iteratively optimization procedure to estimate the number of channels required to reproduce the main characteristics of the macroscopic currents recorded experimentally. A membrane model of the firing of action potentials was then developed based on the kinetic schemes of single channels and the number of channels estimated. We showed that macroscopic currents of human δ-cells can be reproduced by minimal models of single channels when the appropriate number of channels is considered. In addition, our simulations suggest that human δ-cells are capable of generating action potentials through the interaction of the ionic currents involved. Finally, we determined the relative contribution of the currents underlying the firing of action potentials in human pancreatic δ-cells, which allowed us to propose a qualitative model of an action potential in terms of the underlying ionic currents.

Screening models for undiagnosed diabetes in Mexican adults using clinical and self-reported information.

BACKGROUND: Prevalence of diabetes in Mexico has constantly increased since 1993. Since type 2 diabetes may remain undiagnosed for many years, identification of subjects at high risk of diabetes is very important to reduce its impact and to prevent its associated complications. OBJECTIVE: To develop easily implementable screening models to identify subjects with undiagnosed diabetes based on the characteristics of Mexican adults. SUBJECTS AND METHODS: Screening models were developed using datasets from the 2006 and 2012 National Health and Nutrition Surveys (NHNS). Variables used to develop the multivariate logistic regression models were selected using a backward stepwise procedure. Final models were validated using data from the 2000 National Health Survey (NHS). RESULTS: The model based on the 2006 NHNS included age, waist circumference, and systolic blood pressure as explanatory variables, while the model based on the 2012 NHNS included age, waist circumference, height, and family history of diabetes. The sensitivity and specificity values obtained from the external validation procedure were 0.74 and 0.62 (2006 NHNS model) and 0.76 and 0.55 (2012 NHNS model) respectively. CONCLUSIONS: Both models were equally capable of identifying subjects with undiagnosed diabetes (∼75%), and performed satisfactorily when compared to other models developed for other regions or countries.

Modeling Ca(2+) currents and buffered diffusion of Ca(2+) in human ß-cells during voltage clamp experiments.

Macroscopic Ca(2+) currents of the human ß-cells were characterized using the Hodgkin-Huxley formalism. Expressions describing the Ca(2+)-dependent inactivation process of the L-type Ca(2+) channels in terms of the concentration of Ca(2+) were obtained. By coupling the modeled Ca(2+) currents to a three-dimensional model of buffered diffusion of Ca(2+), we simulated the Ca(2+) transients formed in the immediate vicinity of the cell membrane during voltage clamp experiments performed in high buffering conditions. Our modeling approach allowed us to consider the distribution of the Ca(2+) sources over the cell membrane. The effect of exogenous (EGTA) and endogenous Ca(2+) buffers on the temporal course of the Ca(2+) transients was evaluated. We show that despite the high Ca(2+) buffering capacity, nanodomains are formed in the submembrane space, where a peak Ca(2+) concentration between ∼76 and 143 µM was estimated from our simulations. In addition, the contribution of each Ca(2+) current to the formation of the Ca(2+) nanodomains was also addressed. Here we provide a general framework to incorporate the spatial aspects to the models of the pancreatic ß-cell, such as a more detailed and realistic description of Ca(2+) dynamics in response to electrical activity in physiological conditions can be provided by future models.

Mathematical models of electrical activity of the pancreatic ß-cell: a physiological review.

Mathematical modeling of the electrical activity of the pancreatic ß-cell has been extremely important for understanding the cellular mechanisms involved in glucose-stimulated insulin secretion. Several models have been proposed over the last 30 y, growing in complexity as experimental evidence of the cellular mechanisms involved has become available. Almost all the models have been developed based on experimental data from rodents. However, given the many important differences between species, models of human ß-cells have recently been developed. This review summarizes how modeling of ß-cells has evolved, highlighting the proposed physiological mechanisms underlying ß-cell electrical activity.