The modularity of biological regulatory networks

A useful approach to complex regulatory networks consists of modeling their elements and interactions by Boolean equations. In this context, feedback circuits (i.e. circular sequences of interactions) have been shown to play key dynamical roles: whereas positive circuits are able to generate multist ationarity, negative circuits may generate oscillatory behavior. In this pa per, we principally focus on the case of gene networks. These are represent ed by fully connected Boolean networks where each element interacts with al l elements including itself. Flexibility in network design is introduced by the use of Boolean parameters, one associated with each interaction or gro up of interactions affecting a given element. Within this formalism, a feed back circuit will generate its typical dynamical behavior (i.e. multistatio narity or oscillations) only for appropriate values of some of the logical parameters. Whenever it does, we say that the circuit is 'functional'. More interestingly, this formalism allows the computation of the constraints on the logical parameters to have any feedback circuit functional in a networ k. Using this methodology, we found that the fraction of the total number o f consistent combinations of parameter values that make a circuit functiona l decreases geometrically with the circuit length. From a biological point of view, this suggests that regulatory networks could be decomposed into sm all and relatively independent feedback circuits or 'regulatory modules'. ( C) 1999 Elsevier Science Ireland Ltd. All rights reserved.