  Stochastic models for chemical reaction networks have become very popular in
recent years. For such models, the estimation of parameter sensitivities is an
important and challenging problem. Sensitivity values help in analyzing the
network, understanding its robustness properties and also in identifying the
key reactions for a given outcome. Most of the methods that exist in the
literature for the estimation of parameter sensitivities, rely on Monte Carlo
simulations using Gillespie's stochastic simulation algorithm or its variants.
It is well-known that such simulation methods can be prohibitively expensive
when the network contains reactions firing at different time-scales, which is a
feature of many important biochemical networks. For such networks, it is often
possible to exploit the time-scale separation and approximately capture the
original dynamics by simulating a "reduced" model, which is obtained by
eliminating the fast reactions in a certain way. The aim of this paper is to
tie these model reduction techniques with sensitivity analysis. We prove that
under some conditions, the sensitivity values of the reduced model can be used
to approximately recover the sensitivity values for the original model. Through
an example we illustrate how our result can help in sharply reducing the
computational costs for the estimation of parameter sensitivities for reaction
networks with multiple time-scales. To prove our result, we use coupling
arguments based on the random time change representation of Kurtz. We also
exploit certain connections between the distributions of the occupation times
of Markov chains and multi-dimensional wave equations.
