  Finding coarse-grained, low-dimensional descriptions is an important task in
the analysis of complex, stochastic models of gene regulatory networks. This
task involves (a) identifying observables that best describe the state of these
complex systems and (b) characterizing the dynamics of the observables. In a
previous paper [13], we assumed that good observables were known a priori, and
presented an equation-free approach to approximate coarse-grained quantities
(i.e, effective drift and diffusion coefficients) that characterize the
long-time behavior of the observables. Here we use diffusion maps [9] to
extract appropriate observables ("reduction coordinates") in an automated
fashion; these involve the leading eigenvectors of a weighted Laplacian on a
graph constructed from network simulation data. We present lifting and
restriction procedures for translating between physical variables and these
data-based observables. These procedures allow us to perform equation-free
coarse-grained, computations characterizing the long-term dynamics through the
design and processing of short bursts of stochastic simulation initialized at
appropriate values of the data-based observables.
