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.
