A Comparison of Stochastic and Deterministic Model of Insulin Secretion from Islets of Langerhans

Abstract

The main aim of the project is to present a stochastic version of the model of Insulin secretion in islets of Langerhans in pancreatic -cells by Pederson et al.(2009) and account for integral copy numbers of the granules instead of concentrations. The reactions involved in the system corresponding to the granule pools are modelled as a set of coupled ordinary differential equations. We have implemented a hybrid Gillespie stochastic simulation algorithm to produce a stochastic version of this model. In the beginning we implemented the usual Gillespie SSA in order to carry out the stochastic simulations and got discrepancies in comparison to the deterministic solution. As the model of Insulin granule pools contains time-dependent rates we later used a hybrid Gillespie SSA to include time-dependent propensities. The difference in the usual and the hybrid Gillespie algorithm is the step to calculate time of occurrence of the next reaction. Then using the hybrid Gillespie SSA, the average pool sizes were calculated and were compared to the deterministic solution which showed discrepancy in some pools. To check the working and correctness of the algorithm, the algorithm was implemented on related examples and different cases. Euler’s method was used to solve the differential equations involved. For small pool sizes for the IRP chain of the model the deterministic solution were also verified against the solutions using the Master equation. As the discrepancies were more significant in the IRP chain of the model as compared to other pools, cases with different fI(Cmd) functions , number of runs and different Euler time step were tested on the IRP chain. We show the analytical solution for the open and closed systems. Also, we show the mean and variance over stochastic runs for fast and slow depolarisation protocols described by Pederson et al.(2009) matching up with the deterministic solution for the complete model. The calcium compartment functions used are close fits of the Arthur Sherman’s description of the calcium compartment equations. For all the pools, stabilized variance is plotted against mean and deterministic solution choosing random and discrete initial conditions for each run.