  In the past few decades, the development of fluorescent technologies and
microscopic techniques has greatly improved scientists' ability to observe
real-time single-cell activities. In this paper, we consider the filtering
problem associate with these advanced technologies, i.e., how to estimate
latent dynamic states of an intracellular multiscale stochastic reaction
network from time-course measurements of fluorescent reporters. A good solution
to this problem can further improve scientists' ability to extract information
about intracellular systems from time-course experiments.
  A straightforward approach to this filtering problem is to use a particle
filter where particles are generated by simulation of the full model and
weighted according to observations. However, the exact simulation of the full
dynamic model usually takes an impractical amount of computational time and
prevents this type of particle filters from being used for real-time
applications, such as transcription regulation networks. Inspired by the recent
development of hybrid approximations to multiscale chemical reaction networks,
we approach the filtering problem in an alternative way. We first prove that
accurate solutions to the filtering problem can be constructed by solving the
filtering problem for a reduced model that represents the dynamics as a hybrid
process. The model reduction is based on exploiting the time-scale separations
in the original network and, therefore, can greatly reduce the computational
effort required to simulate the dynamics. As a result, we are able to develop
efficient particle filters to solve the filtering problem for the original
model by applying particle filters to the reduced model. We illustrate the
accuracy and the computational efficiency of our approach using several
numerical examples.
