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The CPIM stand for a Lattice model mixing the Contact Process (CP)
https://github.com/jekeymer/Contact-Process
and the Ising Model (IM)
https://en.wikipedia.org/wiki/Ising_model
together in a combined model. The basic idea is to merge both models. Even though Metropolis and Gillespie are not compatible.
The basic idea is to merge both models in order to model the spatial biology of an Ising-like genetic network expressed in a quasi-2D colony of bacteria growing on a surface of solid agar. The genetic network consist of a bi-stable genetic system of reporter genes (RFP vs GFP) coupled by auto-inducer molecules (HSLs)
By fair we mean here using the Gillespie algorithm
https://en.wikipedia.org/wiki/Gillespie_algorithm
in the Monte Carlo method updating the Lattice. This is a fundamental requirement as there are Lattice states with two possible reactions. Of these only differentiation is commensurable as it does represent a rate. Metropolis flips are not.
Lattice states are: 0: vacancy, representing an empty site suitable for colonization by its neighbours (plotted in black) 2: undifferentiated state (plotted in white) -1: spin down, representing a site expressing one reporter gene (plotted in green) +1: spin up, representing a site expressing the other reporter (plotted in magenta)
Occupied sites (states: -1,2,+1) can have spin flips or differentiation reactions together with particle death reactions.
So we use only Gillespie (version Partial Gillespie) for the differentiation reaction function update_lattice_2 For the spin flip, we only do it if the cell survives the time step. Thus no Gillespie.
To compare (version full Gillespie) what happens otherwise, we use Gillespie for flips and see that the Metropolis Algorithm is not compatible with Gillespie as it is not suppose to represent rates. We see then (if ran) it affects the contact process death rate in an artificial fashion. For this test we made the function: update_lattice_1
In the Gillespie algorithm we use 2 random numbers. The first one is used to decide which one of the competing reactions might take place. Then, a second random number is used to decide if the chosen reaction, indeed take place (or not). This way, the Poisson process is respected and probabilities are well defined.