Dynamic modularity in discrete-time models of regulatory networks

Highlights: ► We study modules in biological regulatory networks as non-autonomous dynamical systems. ► We show how a module adapts to different functionalities depending on the environment. ► We investigate the implications on the design of regulatory networks from such modules. - Abstract: We study the properties of small regulatory networks treated as non-autonomous dynamical systems, otherwise called modules when working inside larger networks. We explicit and study the conditions on the input sequences and the internal parameters of the system to behave as a transducer (finite-state automata with inputs and outputs). In the allowed families of networks, we distinguish those with and without feedback on the basis of whether the internal dynamics of the module has a role on determining their input–output behaviors or not. The input–output and non-autonomous bifurcation analysis of this class of modules rely on studying their symbolic dynamics. We consider the interplay between the internal and structural properties of the modules and the different possible inputs on them to deduce possible new functionalities as internal and external responses. Far from the over-optimistic view according to which to a module shall correspond one functionality, we obtain a trade-off between a large spectrum of behaviors and the robustness of each of them depending on the delays, non-linearities and strengths involved in the regulations.