The effect of unhealthy β-cells on insulin secretion in pancreatic islets
© Pu et al; licensee BioMed Central Ltd. 2013
Published: 11 November 2013
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© Pu et al; licensee BioMed Central Ltd. 2013
Published: 11 November 2013
Insulin secreted by pancreatic islet β-cells is the principal regulating hormone of glucose metabolism and plays a key role in controlling glucose level in blood. Impairment of the pancreatic islet function may cause glucose to accumulate in blood, and result in diabetes mellitus. Recent studies have shown that mitochondrial dysfunction has a strong negative effect on insulin secretion.
In order to study the cause of dysfunction of pancreatic islets, a multiple cell model containing healthy and unhealthy cells is proposed based on an existing single cell model. A parameter that represents the function of mitochondria is modified for unhealthy cells. A 3-D hexagonal lattice structure is used to model the spatial differences among β-cells in a pancreatic islet. The β-cells in the model are connected through direct electrical connections between neighboring β-cells.
The simulation results show that the low ratio of total mitochondrial volume over cytoplasm volume per β-cell is a main reason that causes some mitochondria to lose their function. The results also show that the overall insulin secretion will be seriously disrupted when more than 15% of the β-cells in pancreatic islets become unhealthy.
Analysis of the model shows that the insulin secretion can be reinstated by increasing the glucokinase level. This new discovery sheds light on antidiabetic medication.
With 25-30 percent of adults in the developed world having a high risk of diabetes, and 24 million people in the United States with diabetes, diabetes mellitus ranks as a principal cause of death . The incidence of diabetes mellitus is expected to double within the coming 20 years. After a meal, healthy individuals secrete insulin into the bloodstream signaling the consumption of glucose produced from the digested food. Thus insulin has an important signaling role in the maintenance of low blood glucose levels via glucose consumption. Type II diabetes may result from disruption of this signaling process [2, 3]. Hence understanding the insulin secretion mechanism is vitally important.
The cells, in fact, have differing capabilities in consuming glucose, since their differing physical properties cause different rates of electron transport chain function, ATP production, Ca2+ uptake and release, and cell membrane potential. Furthermore, some β-cells do not function normally as just described. The term unhealthy is used to describe cells that cannot consume glucose normally. Unhealthy cells interact strongly with adjacent healthy cells, because of their electrical connections. The total insulin secretion is the aggregation of the insulin secreted by all pancreatic islet cells. Its dynamics may be destroyed by a small fraction of unhealthy cells even though a large majority of the cells are still healthy. Consequently, understanding how a cohort of unhealthy cells can affect the insulin secretion process in other healthy cells is very important, and can lead to a better understanding of the cause of diabetes.
This present work proposes a mathematical model to study scenarios with unhealthy cells leading to insulin oscillation failure. The new model is based on a mathematical insulin secretion model of the oscillation and metabolic pathway of insulin secretion in a single β-cell, due to Bertram et al. [16–18]. Bertram's original model is extended to model spatial and coupling effects  due to the electrical connections between neighboring β-cells. Mitochondrial dysfunction is believed to have a strong negative effect on insulin secretion [19–21], and also mitochondrial dysfunction is believed to be a main cause of unhealthy cells, based on the recent data of [22–24]. The ratio of the total mitochondrial volume to the cytoplasm volume per β-cell, a parameter corresponding to mitochondria function, is used to define unhealthy cells, as distinguished from healthy cells. The effect on the islet insulin oscillation of unhealthy cells coupled with healthy cells is then studied in detail. Multiple cell simulations demonstrate insulin oscillation malfunction when the fraction of unhealthy cells exceeds approximately 15%. To verify this result, a simplified coupling topology is used to study the effect of one unhealthy cell on neighboring healthy cells. The latter study confirmed the previous more complicated simulation.
In order to understand the cause of insulin secretion failure resulted from unhealthy cells, an eight-cell model is built and studied in details. We discovered a critical difference of the dynamics between healthy cells and unhealthy cells. Our simulation results demonstrate that stimulating glucokinase can make unhealthy pancreatic islets function normally. Based on the discovery, a possible strategy for antidiabetic medicine is proposed. Our strategy is consistent with recent antidiabetic medicine development [25–30] that identifies glucokinase as a major drug target.
where I K , I Ca , I K(Ca), and I K(ATP) are the ionic current on the membrane, C is the membrane capacitance, f c is the fraction of free Ca2+ in the cytosol, J mem is the flux of Ca2+ across the plasma membrane, J m is the flux of Ca2+ out of the mitochondria, which is scaled by the mitochondria/cytosol volume ratio κ, J er is the flux out of the ER, f er is the fraction of free Ca2+ in the ER, V c and V er are the volumes of the cytosolic and ER compartments, respectively, n is the fraction of the open delayed rectifying K+ channels, τ is the relaxation time for the open and close reactions of the delayed rectifying K+ channels to reach equilibrium, and the steady state function for n is n ∞(V) = (1 + exp(−(V + 16)/5))−1. For more details on these terms in the differential equations of the model, refer to Bertram et al.
The coordinates of the neighbors of the cell with coordinates (a, b, c)
Coordinates (x, y, z)
(a − 1, b, c)
(a − 1, b + 1, c + 1)
(a − 1, b, c + 1)
(a, b, c − 1)
(a, b, c + 1)
(a, b − 1, c − 1)
(a, b − 1, c)
(a, b + 1, c)
(a, b + 1, c + 1)
(a + 1, b − 1, c − 1)
(a + 1, b, c − 1)
(a + 1, b, c)
Besides the spatial differences, each cell has its own parameter set. Among all the parameters of the single cell model, κ, the ratio of mitochondria volume to cytosol volume, represents the function of mitochondria. The single cell study has shown that the oscillation behavior of a cell is very sensitive to κ, and thus κ is used to describe the cell differences. The value of κ is different for each cell in the 3-D structure, while all other cell parameters are set the same.
The differential equations for the model, implemented in Fortran 90, were solved using LSODE in ODEPACK . Generated data were plotted using Matlab. The simulations are performed on a Mac OS 2.4 GHz Intel(R) Core 2 Duo CPU with 4GB memory.
where J hyd is the hydrolysis rate of ATP to ADP, κ (κ = 0.0733 in the standard model ) is the ratio of the total functional mitochondrial volume to the cytoplasm volume, and J ANT is the flux through the adenine nucleotide translocator, which exchanges ADP and ATP molecules between the cytoplasm and the mitochondria. With κ = 0.0733, both the cell membrane potential and insulin secretion show periodic bursts of rapid oscillation, as illustrated in Figure 1 of Additional file 1. The slowest component of the compound bursting is due to oscillatory glycolysis, reflected by an oscillatory FBP concentration. The oscillatory glycolysis causes slow oscillations in ADP, which superimpose with the faster ADP oscillations driven by Ca2+. The multimodal ADP rhythm leads to oscillations in the conductance of the ATP-dependent potassium channel, which drives the burst episodes of the membrane potential V, which then results in compound bursting of intracellular calcium, leading to pulsatile insulin secretion.
In truth, though, κ varies between cells; specifically, cells with some mitochondria weakened due to aging will effectively have a κ smaller than the default value. It is therefore paramount to study the sensitivity of the cell membrane potential V and insulin secretion I patterns to the parameter κ. For this work, Bertram's single cell model is modified by varying the parameter κ to represent different levels of functional mitochondria within different cells. In terms of changes in the patterns of V, Ca2+, and I, the bursting behavior of the model is classified into four categories as a function of the volume of functioning mitochondria in the β-cell, in order of occurrence: burst formation, periodic burst, burst loss, and decoupling.
The bursting pattern starts to form at around 85% of the regular mitochondrial volume, which is when the total volume of the mitochondria is about 6% of the cytoplasmic volume. If the volume of functioning mitochondria is lower than this limit, insufficient ATP is produced to allow the insulin burst. In the burst formation category (Additional file 1: Figure 2) each burst consists of a small number of spikes in plasma membrane voltage, cytoplasmic calcium ion concentration, and the insulin secretion rate. If the mitochondrial volume fraction is increased beyond the value used in the standard model, the behavior of the burst begins to change in pattern, entering the burst loss category (Additional file 1: Figure 3). In the burst loss category every other burst decreases in duration until it is lost completely, while the remaining bursts lengthen their duration. The net effect is a decrease in the fraction of time spent in each burst.
In order to quantify the development and eventual loss of the bursting behavior, define the "bursting fraction" as the portion of the total time taken up by the bursts, where a burst duration is defined as the time from the first to the last peak in a burst. Bertram's single cell original model has a bursting fraction of approximately 0.55 (meaning that a burst in the model variables' oscillations is occurring for 55% of the time). When the cell mitochondrial volume fraction is changed, this bursting fraction changes markedly (cf. Figure 4), and the patterns of the calcium ion concentration Ca2+, the cell membrane potential V, and the insulin secretion rate I all alter noticeably. Figure 4 shows the ranges of the mitochondrial volume ratios corresponding to the four classifications: burst formation, periodic bursts, burst loss, and decoupling. Since the classifications are qualitative, the divisions between the classifications are blurred and the ranges are drawn overlapping. The membrane voltage, calcium ion concentration, and insulin release all have nearly identical bursting fractions for the first three categories (formation, periodic busts, and loss). In the decoupling category the bursting fractions in the calcium ion concentration and insulin release rapidly fall to zero, while the voltage bursting fraction holds relatively constant. The normal bursting behavior of the single β-cell model occurs only within a narrow range of mitochondrial volumes, from roughly 7% to 8% of the cellular volume.
The insulin bursts are totally absent when the volume fraction of the mitochondria is either less than 0.06 or greater than 0.095, which suggests an alternative definition for "healthy cell" and "unhealthy cell". Define a healthy cell as having a mitochondria volume fraction κ between 0.06 and 0.095, and a cell with κ out of this range as an unhealthy cell. It is plausible that some mitochondria lose their function, effectively resulting in a smaller κ. The ensuing numerical experiments use κ = 0.05 for unhealthy cells and the standard value κ = 0.0733 for healthy cells.
Although there are about 1,000 cells in each pancreatic islet, for multiple β-cells simulations, consider first the case with 125 cells coupled spatially in a 3-D hexagonal lattice. The justification for using 125 rather than 1000 cells is a pragmatic one--the CPU time for simulating 1,000 cells is rather long. In a 1,000 cell heterogeneous model, each single cell model has ten variables, yielding a 10,000-dimensional ODE, which required 26 hours to solve till time t = 2 × 106. Furthermore, the oscillation patterns observed from 125 cells are qualitatively very similar to those observed from 1,000 cells.
If there are no unhealthy cells at all, the membrane potentials of all the cells synchronize after the coupling is turned on at time 400,000 milliseconds (ms) by changing gc from 0 to 150. This simulation of 125 cells without unhealthy cells is shown in Figures 1 (membrane potential) and 2 (total insulin secretion) in Additional file 2. The curves in the membrane potential plot are out of phase at time t = 0, but soon after 400,000 ms, these curves coalesce (see Additional file 2: Figure 1). Because the insulin levels of some cells are high while those of other cells are low, the total insulin is relatively flat before synchronization. Immediately after the coupling is turned on, the total insulin secretion shows bursts and its value rises to a hundred times that of a single cell, because there are more than a hundred cells synchronized and releasing insulin in phase.
Focus on total insulin secretion to see how unhealthy cells, through the 3-D coupling in the hexagon structure, affect the total insulin secretion. To save computational time the coupling is turned on at the beginning of the simulations, t = 0. Figure 3 in Additional file 2 shows the resulting total insulin behavior with 10% of the cells being unhealthy spread uniformly in the 3-D hexagonal structure. The total insulin, as in the case of 100% healthy cells, shows periodic oscillations and maintains a reasonable level. When the percent of cells being unhealthy increases to 15%, the oscillations of total insulin still look normal (see Additional file 2: Figure 4), but now some bursts have fewer spikes. As the percentage of cells being unhealthy increases to 20% and 30% from 10% and 15% of cells being unhealthy, the spikes within each burst become much less numerous (shown in Additional file 2: Figures 5 and 6). These bursts are also much more irregular, and even more significantly, totally disappear after 2.25 × 106 ms (Additional file 2: Figure 6). In summary, the cohort of unhealthy cells dominates the global behavior, resulting in a level of total insulin too low to maintain proper pancreatic islet function. The conclusion is that if there is more than approximately 15% of cells being unhealthy, the function of the pancreatic islet will be severely affected.
To verify the conclusions from simulations with 125 cells, simulations were also performed for the model with 1,000 cells. This 1000 cell model cannot be run to as long a time as the 125 cell model, but observe that when, for a certain time, insulin secretion cannot generate enough spikes, the pancreatic islet can be considered to be malfunctioning. Hence the simulation monitors the numbers of spikes in the last five bursts, and if the mean number of spikes is below three, deems that the overall system is malfunctioning and halts. The simulations for 1,000 cells with 30%, 20%, 15% of cells being unhealthy, all considered malfunctioning systems, are shown in Figures 7, 8, and 9 of Additional file 2 respectively. A simulation for 10% of cells being unhealthy found no malfunction in an extremely long time (comparable to the time for the 125 cell model runs). In summary, the 125 cell and 1000 cell simulations yield similar conclusions: If the percentage of cells being unhealthy is larger than approximately 15%, the system will malfunction; at 15% of cells being unhealthy, the system still functions but is close to malfunctioning; below 10% of cells being unhealthy, the system can definitely function very well.
Comparison of Two Different Ratios
Ratio of Links
Ratio of Cells
Since an unhealthy cell has a smaller mitochondria/cytosol volume ratio than that of a healthy cell, unhealthy cells usually have lower ATP (from oxidative phosphorylation) levels. ATP has a negative feedback to phosphofructokinase (PFK) in glycolysis. When the ATP level is lower, the PFK reaction becomes faster, which directly consumes G6P faster leading to lower G6P levels in unhealthy cells than in healthy cells. As shown in Figure 10 of Additional file 4 the levels of G6P in healthy cells are higher than those in unhealthy cells. At the same time, because of the low production rate of ATP in unhealthy cells, the ratio of ATP to ADP gets lower. ATP-sensitive K+ channels in the plasma membrane are activated by ADP and inactivated by ATP, so the ratio of these nucleotides determines the fraction of open ATP-sensitive K+ channels. When the ATP/ADP ratio is low there is an increase in the number of open ATP-sensitive K+ channels, which results in the difficulty of membrane depolarization. Voltage-dependent Ca2+ channels are blocked. Since the insulin is secreted when Ca2+ exceeds a certain level, the blocked Ca2+ channels will reduce insulin secretion. Therefore, raising the levels of G6P in unhealthy cells back to normal will bring back normal insulin secretion. This analysis suggests the hypothesis that increasing the value of any substance ahead of the ATP synthesis in the pathway, such as the substances FBP and NADH in mitochondria, reinstates the oscillations. Manually resetting the values of either [FBP] or [NADH] in all the cells to the same initial value confirms the conjecture: not only can G6P restart the insulin oscillations, but also FBP and NADHm can restart the oscillations.
Since G6P and FBP are both substances in the glycolysis pathway, if the rate of glycolysis were faster, the oscillations of insulin might be resumed as well. In order to test this conjecture, the glucokinase level is raised fourfold after the insulin failure is detected. Glucokinase is the enzyme that phosphorylates glucose to glucose-6-phosphate (G6P). Figures 3 and 4 in Additional file 5 show that the oscillations of insulin are resumed when the glucokinase level increased. Two oscillation periods are shown in Figure 5 of Additional file 5 (from the earlier 125 cell model): The longer period (slow oscillations) is caused by the glycolytic oscillations, while Ca2+ feedback is responsible for the fast oscillations. Only the short period remains in Figure 6 of Additional file 5. This observation suggests that if the longer period could be reestablished in Figure 6 of Additional file 5 the insulin oscillations might be reinstated. Since the glycolysis pathway is responsible for the slow oscillations, that pathway may need some stimulus. The simulation results in Figures 4 and 5 of Additional file 5 demonstrate that by increasing the glucokinase level in the glycolysis pathway, the insulin oscillations can be resumed after the failure.
In conclusion, the insulin secretion of pancreatic islets usually will be functionally destroyed when there are more than 20% of cells being unhealthy among all cells. The more unhealthy cells there are, the more irregular insulin secretion will be. Increasing the level of glucokinase can make the pancreatic islet function normally when there is a high fraction of unhealthy cells by increasing the glucose absorption of the glycolysis pathway. This has implications for the clinical treatment of type II diabetes. Currently there are three classes of medications used to treat type II diabetes. The first treatment is to increase the amount of insulin secreted by the pancreas by inhibiting the opened delayed rectifying K+ channels. The second treatment is to increase the sensitivity of target organs to insulin. The third treatment is to decrease the rate at which glucose is absorbed from the gastrointestinal tact, which is a method to reduce the glucose uptake from food. It appears that all the available treatments are insufficient to stem the tide. Therefore, new treatments are currently under investigation including the development of therapeutic agents with novel action mechanisms. Recently, researchers have identified glucokinase as an outstanding drug target for developing antidiabetic medicines [25–30]. Assuming the mathematical models are valid, the simulation results demonstrate that stimulating glucokinase can make unhealthy pancreatic islets function normally, consistent with new antidiabetic medicines. Such multiple cell models are good candidates for guiding the development of the next generation of antidiabetic medicines.
flavin adenine dinucleotide
glyceradehyde 3-P dehydrogenase
nicotinamide adenine dinucleotide plus hydrogen
permeability transition pore
This work is based on the conference paper "The effect of unhealthy beta-cells in synchronized insulin secretion", which appeared in the 2012 IEEE International Conference on Bioinformatics and Biomedicine (BIBM 2012) . This work was partially supported by the National Science Foundation under awards CCF-0726763 and CCF-0953590, and the National Institutes of Health under award GM078989.
The publication costs for this article were funded by the corresponding author with the above sources of support.
This article has been published as part of BMC Medical Genomics Volume 6 Supplement 3, 2013: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine 2012: Medical Genomics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcmedgenomics/supplements/6/S3.
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