  It is well known that the kinetics of an intracellular biochemical network is
stochastic. This is due to intrinsic noise arising from the random timing of
biochemical reactions in the network as well as due to extrinsic noise stemming
from the interaction of unknown molecular components with the network and from
the cell's changing environment. While there are many methods to study the
effect of intrinsic noise on the system dynamics, few exist to study the
influence of both types of noise. Here we show how one can extend the
conventional linear-noise approximation to allow for the rapid evaluation of
the molecule numbers statistics of a biochemical network influenced by
intrinsic noise and by slow lognormally distributed extrinsic noise. The theory
is applied to simple models of gene regulatory networks and its validity
confirmed by comparison with exact stochastic simulations. In particular we
show how extrinsic noise modifies the dependence of the variance of the
molecule number fluctuations on the rate constants, the mutual information
between input and output signalling molecules and the robustness of
feed-forward loop motifs.
