Introduction

Langerhans islets in pancreas produces bursts of electrical activity when the beta cells are exposed to stimulatory blood glucose levels. These bursts are usually of a regular repeating pattern with a period of 10s to 1 minute. However, under certain circumstances, the bursts appear in clusters of bursts of bursts which is nominated as compound bursting. Along with other response factors of pancreatic beta cells including Ca concentration, oxygen consumption, membrane potential and glucose levels(intraislet), the compound bursting exhibits slow oscillations as well as superimposed rapid oscillations. The most rapid component of the oscillations are in phase with oscillations of free Ca2+ concentration in pancreatic B cells; the second component with a period of 5-10 minutes has a physiological role, and is missing from type II diabetes patients; slower ultradian rhythms with a period of 2 hours[2] and circadian rhythms are also observed[3]. To study the general pattern of pancreatic beta cell bursts along with the different modes of bursting, Bertram et al. (2004) [1] introduced this multi-component mathematical model, mainly focusing on the 2 faster components of the oscillatory secretion of insulin.

The Model

The authors have constructed a simplified and schematic ODE model of the general pattern and different components of the multi-model oscillation of pancreatic beta cell bursts. The model is focused on four fundamental components of the burst: the electrical currents, the flow of calcium, the concentration of adenine nucleotides, and glycolysis. These components of the beta cell burst oscillate with a shared ground frequency, but with different pattern in both oscillation mode and intensity. The model simulations are mainly focusing on 4 representative entities or measurements of each component, which are the membrane potential, the intracellular calcium concentration, ADP and Fructose-1,6-bisphosphate(FBP). The overall mechanism of glycolysis and its components in the model are shown in figure 1. The model includes 7 ordinary differential equations expressing the level of molecules, currents and membrane potential in the model. There are 113 parameters including 17 species which can be expressed in formulas involving the modelled species with ODE in this model.

Figure 1. Schematic representation of glycolysis.

Results

The authors fitted the parameters obtained by estimation. This allowed them to simulate the time-dependent evolution of the model entities during the burst. The results of compound bursting are shown in figure 2. The voltage has shown as a compound bursting pattern, parallel to calcium concentration; the slow, large rhythm of oscillation is due to the glycolytic system, while ADP oscillated in a similar pattern, but with small deflections out of phase with Ca2+. This is due to the negative effects of Ca2+ on respiration. Simulations of the oscillation driven by individual systems are done; a separate simulation of slow glycolytic and subthreshold oscillations is also performed. Some relevant observations are as follows:

1. Bursting can occur without glycolytic oscillations;

2. Glycolytic oscillations convert regular bursting to compound bursting, as its slow mode

3. Glycolytic oscillations can also drive slow glycolytic bursting and subthreshold oscillations

4. ADP is out of phase with Ca2+ during glycolytic oscillations, and is in phase during Ca2+ driven bursting.