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Models of reaction networks within interacting compartments (RNIC) are a generalization of stochastic reaction networks. It is most natural to think of the interacting compartments as “cells” that can appear, degrade, split, and even merge, with each cell containing an evolving copy of the underlying stochastic reaction network. Such models have a number of parameters, including those associated with the internal chemical model and those associated with the compartment interactions, and it is natural to want efficient computational methods for the numerical estimation of sensitivities of model statistics with respect to these parameters. Motivated by the extensive work on computational methods for parametric sensitivity analysis in the context of stochastic reaction networks over the past few decades, we provide a number of methods in the basic RNIC setting. Provided methods include the (unbiased) Girsanov transformation method (also called the likelihood ratio method) and a number of coupling methods for the implementation of finite differences, each motivated by methods from previous work related to stochastic reaction networks. We provide several numerical examples comparing the various methods in the new setting. We find that the relative performance of each method is in line with its analog in the “standard” stochastic reaction network setting. We have made all the MATLAB codes used to implement the various methods freely available for download. 

Stochastic reaction networks, which are usually modeled as continuous-time Markov chains on $$\ZZ^d_{\ge 0}$$, and simulated via a version of the ``Gillespie algorithm,'' have proven to be a useful tool for the understanding of processes, chemical and otherwise, in homogeneous environments. There are multiple avenues for generalizing away from the assumption that the environment is homogeneous, with the proper modeling choice dependent upon the context of the problem being considered. One such generalization was recently introduced in \cite{Duso_Zechner_2020}, where the proposed model includes a varying number of interacting compartments, or cells, each of which contains an evolving copy of the stochastic reaction system. The novelty of the model is that these compartments also interact via the merging of two compartments (including their contents), the splitting of one compartment into two, and the appearance and destruction of compartments. In this paper we begin a systematic exploration of the mathematical properties of this model. We (i) obtain basic/foundational results pertaining to explosivity, transience, recurrence, and positive recurrence of the model, (ii) explore a number of examples demonstrating some possible non-intuitive behaviors of the model, and (iii) identify the limiting distribution of the model in a special case that generalizes three formulas from an example in \cite{Duso_Zechner_2020}.