  Here we propose a new approach to modeling gene expression based on the
theory of random dynamical systems (RDS) that provides a general coupling
prescription between the nodes of any given regulatory network given the
dynamics of each node is modeled by a RDS. The main virtues of this approach
are the following: (i) it provides a natural way to obtain arbitrarily large
networks by coupling together simple basic pieces, thus revealing the
modularity of regulatory networks; (ii) the assumptions about the stochastic
processes used in the modeling are fairly general, in the sense that the only
requirement is stationarity; (iii) there is a well developed mathematical
theory, which is a blend of smooth dynamical systems theory, ergodic theory and
stochastic analysis that allows one to extract relevant dynamical and
statistical information without solving the system; (iv) one may obtain the
classical rate equations form the corresponding stochastic version by averaging
the dynamic random variables (small noise limit). It is important to emphasize
that unlike the deterministic case, where coupling two equations is a trivial
matter, coupling two RDS is non-trivial, specially in our case, where the
coupling is performed between a state variable of one gene and the switching
stochastic process of another gene and, hence, it is not a priori true that the
resulting coupled system will satisfy the definition of a random dynamical
system. We shall provide the necessary arguments that ensure that our coupling
prescription does indeed furnish a coupled regulatory network of random
dynamical systems. Finally, the fact that classical rate equations are the
small noise limit of our stochastic model ensures that any validation or
prediction made on the basis of the classical theory is also a validation or
prediction of our model.
