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Modeling: Insulin Secretion
Abstract
In this model, we simulate the insulin secretion behavior of pancreatic β cells, trying to realize the similar curve in a single β cell, as that of glucose-insulin secretion which has been shown in literature, according to the reported insulin secretion mechanisms.
In general, we divide the process from glucose sensation to insulin secretion into three sessions. The first session covers the metabolic process from glucose intake to production of ATP. The second session covers the behavior of ion channels after the change in [ATP]/[ADP] ratio. The last session depicts the insulin secretion activated by cellular [Ca2+] rise.
This single-cell model is key to the overall insulin secretion curve due to the synchronization of the cells in the same pancreas islet.
Introduction
In physiology, the secretion of insulin is tightly regulated. After food intake and rising of blood glucose concentration, insulin is quickly secreted to down tune the blood glucose. Lack of insulin as well as its overproduction causes harm to human bodies. Thus it is really a need to determine the physiological parameters during insulin secretion. Of the first priority is the [glucose]-insulin secretion curve.
Fig. 1 Insulin-related metabolic pathways in β cells (adapted from [1])
Task Analysis
Our model is in fact a time-amount model addressing the total insulin secretion amount with time under given glucose concentration.
The secretion of insulin is considered as the result of a series of regulation, which includes acceleration of metabolism, opening of ATP-sensitive K+ channel, depolarization of membrane, influx of Ca2+, etc. While this is not the whole picture, there is another pathway leading to the secretion of insulin, as is shown in the graph above, but due to its complexity and lack of studies, we focus on the major and clearer part of the pathway.
There are two phases of insulin secretion. First of them comes from the drastic reaction to glucose of β cells in all islets, and the second comes from the difference of secretion paces of different islets.
The main events are shown in the following chart:
[Glc] rising → [ATP]/[ADP] change → KATP Ch closure
→ Ca2+ Ch opening → [Ca2+]c rising → insulin secretion
As the insulin secretion-[Ca2+] is quite expectable, we focus mainly on the [Ca2+]-time curve.
Modeling
We make the following assumptions:
- The reactions in metabolic process obey the M-M equations;
- There is a time interval between [ATP[/[ADP] ratio reaches the threshold and closure of K+ ch;
- The infux of Ca2+ is only subject to the chemical and electrical gradient, and cellular retrieval of Ca2+ is constant.
Stage I
Michaelis-Menten equation:
Normally [ATP]/[ADP]= x0 (during fast)
X1(t) = 
Where δx(t) = 
∈(0,1), p is the concentration of ATP, and [G] is that of the glucose.
We assume that under discrete time intervals reactions conduct in packet 
.
Stage II
K+ Channel opening state:
K(t) = 
K(t) limiting conditions:
K(0) = 0;
K(t) = 1, when X1(t) ≥ Ra;
K(t) = 0, when X1(t) ≤ Ra;
Amount of imported Ca2+:
Suppose the intracellular Ca2+ concentration is:
(*)
Note 1 for the successful application of iteration, we introduce F(t) and C(t) with relationship as:
F(i) = exp(-c1)*(F(i-1)-1) + 1;
C(i) = c*(F(i) - F(i-1)) - c2 + C(i-1);
These two functions in fact realize the behavior of the above C(t).
Note 2 we assume that there should be a time period after [ATP]/[ADP] reaches the threshold before the K+ channel closes, resulting in a time delay:
if i > 150
for j = 1:150
if K(i-j) == 1
q = 1;
end
end
end
Stage III: Ca2+-induced insulin secretion
With the increase in cellular [Ca2+] concentration, insulin secretion comes into a steady state concerned with [Ca2+] after an intense release.
I(t) = 
Thus we are able to get an insulin secretion curve.
Results and Analysis
Our program gives the following [Ca2+]-t graph:
Fig. 2 simulated [Ca2+]-t curve (x-axis in unit of 0.01min)
Physiological curve as follows:
Fig. 3 physiological insulin secretion curve
APPENDIX: MATLAB code
To read detailed code, please visit:
https://github.com/JwadeWayne/Biologeek/blob/master/Insulin%20Release
Reference
[1] J. C. Henquin, Regulation of insulin secretion: a matter of phase control and amplitude modulation, Diabetologia (2009) 52:739–751.
[2] Evren U. Azeloglu, et al., Interconnected Network Motifs, Control Podocyte Morphology and Kidney Function, Science Signaling (311), ra12.
[3] Lehninger Principles of Biochemistry 5th ed., David L. Nelson, Michael M. Cox, W. H. Freeman and Company, New York, 2007.
[4] Physiology first edition, Yao Tai, et al., People’s Hygiene Press, 2008.11.
[5] Philippe D.et al., Interplay between cytoplasmic Ca2+ and the ATP/ADP ratio: a feedback control mechanism in mouse pancreatic islets. Biochem. J. (1998)333, 269–274.
[6] Barbara E. et al., Regulation of Steady-state Free Ca2+ Levels by the ATP/ADP Ratio and Orthophosphate in Permeabilized RINm5F Insulinoma Cells, The Journal of Biological Chemistry. 1988, Vol 263, No.9:4247-4253.
