  Cellular processes do not follow deterministic rules; even in identical
environments genetically identical cells can make random choices leading to
different phenotypes. This randomness originates from fluctuations present in
the biomolecular interaction networks. Most previous work has been focused on
the intrinsic noise (IN) of these networks. Yet, especially for
high-copy-number biomolecules, extrinsic or environmental noise (EN) has been
experimentally shown to dominate the variation. Here, we develop an analytical
formalism that allows for calculation of the effect of EN on gene-expression
motifs. We introduce a method for modeling bounded EN as an auxiliary species
in the master equation. The method is fully generic and is not limited to
systems with small EN magnitudes. We focus our study on motifs that can be
viewed as the building blocks of genetic switches: a nonregulated gene, a
self-inhibiting gene, and a self-promoting gene. The role of the EN properties
(magnitude, correlation time, and distribution) on the statistics of interest
are systematically investigated, and the effect of fluctuations in different
reaction rates is compared. Due to its analytical nature, our formalism can be
used to quantify the effect of EN on the dynamics of biochemical networks and
can also be used to improve the interpretation of data from single-cell
gene-expression experiments.
