Dynamical robustness of biological networks with hierarchical distribution of time scales.

Concepts of distributed robustness and r-robustness proposed by biologists to explain a variety of stability phenomena in molecular biology are analysed. Then, the robustness of the relaxation time using a chemical reaction description of genetic and signalling networks is discussed. First, the following result for linear networks is obtained: for large multiscale systems with hierarchical distribution of time scales, the variance of the inverse relaxation time (as well as the variance of the stationary rate) is much lower than the variance of the separate constants. Moreover, it can tend to 0 faster than 1/n, where n is the number of reactions. Similar phenomena are valid in the nonlinear case as well. As a numerical illustration, a model of signalling network is used for the important transcription factor NFkappaB.