A Model of Gene Expression Based on Random Dynamical Systems Reveals Modularity Properties of Gene Regulatory Networks

摘要

Here we propose a new approach to modeling gene expression based on thetheory of random dynamical systems (RDS) that provides a general couplingprescription between the nodes of any given regulatory network given thedynamics of each node is modeled by a RDS. The main virtues of this approachare the following: (i) it provides a natural way to obtain arbitrarily largenetworks by coupling together simple basic pieces, thus revealing themodularity of regulatory networks; (ii) the assumptions about the stochasticprocesses used in the modeling are fairly general, in the sense that the onlyrequirement is stationarity; (iii) there is a well developed mathematicaltheory, which is a blend of smooth dynamical systems theory, ergodic theory andstochastic analysis that allows one to extract relevant dynamical andstatistical information without solving the system; (iv) one may obtain theclassical rate equations form the corresponding stochastic version by averagingthe dynamic random variables (small noise limit). It is important to emphasizethat unlike the deterministic case, where coupling two equations is a trivialmatter, coupling two RDS is non-trivial, specially in our case, where thecoupling is performed between a state variable of one gene and the switchingstochastic process of another gene and, hence, it is not a priori true that theresulting coupled system will satisfy the definition of a random dynamicalsystem. We shall provide the necessary arguments that ensure that our couplingprescription does indeed furnish a coupled regulatory network of randomdynamical systems. Finally, the fact that classical rate equations are thesmall noise limit of our stochastic model ensures that any validation orprediction made on the basis of the classical theory is also a validation orprediction of our model.