Analyzing fixed points of intracellular regulation networks with interrelated feedback topology

Here we extend previous work and show several connections between the circuit-characteristic and the stability of fixed points. In particular, we derive a sufficient condition on the characteristic for a fixed point to be unstable for certain graph structures and demonstrate that the characteristic does not contain the information to decide whether a fixed point is asymptotically stable. All statements are illustrated on biological network models.Single feedback circuits and their role for complex dynamic behavior of biological networks have extensively been investigated, but a transfer of most of these concepts to more complex topologies is difficult. In this context, our algorithm is a powerful new approach for the analysis of regulation networks that goes beyond single isolated feedback circuits.Describing the dynamic behavior of molecular interactions in a cell or cell compartment by chemical reaction kinetics has become a standard approach in systems biology for metabolic pathways as well as for regulatory networks. Since qualitative knowledge about these interactions is often available from experiments, literature or databases, which can be represented as network graphs, several different graph-based approaches have been developed to analyze the behavior of the networks. These methods operate solely on the graphs without detailed knowledge of the kinetic rates. They show for example that certain subnetwork structures are necessary to generate complex behavior such as oscillations, hysteresis or multistationarity. Thus, such behavior can be excluded for relatively small and simple networks that lack these subnetworks. So far, most of these approaches have the following limitations for practical use: First, they only allow to make statements for relatively simple graph topologies, and second, they are often restricted to very specific model classes such as metabolic networks of the form x ？ = Sv ( x ) with stoichiometric matrix S and (often polynomial) flux