Analyzing steady states of dynamics of bio-molecules from the structure of regulatory networks

Regulatory relations between biological molecules constitute complex network systems and realize diverse biological functions through the dynamics of molecular activities. However, we currently have very little understanding of the relationship between the structure of a regulatory network and its dynamical properties. In this paper we introduce a new method, named "linkage logic" to analyze the dynamics of network systems. By this method, we can restrict possible steady states of a given complex network system from the knowledge of regulatory linkages alone. The regulatory linkage simply specifies the list of variables that affect the dynamics of each variable. We formalize two aspects of the linkage logic: the "Principle of Compatibility" determines the upper limit of the diversity of possible steady states of the dynamics realized by a given network; the "Principle of Dependency" determines the possible combinations of states of the system. By combining these two aspects, (i) for a given network, we can identify a cluster of nodes that gives an alternative representation of the steady states of the whole system, (ii) we can reduce a given complex network into a simpler one without loss of the ability to generate the diversity of steady states, (iii) we can examine the consistency between the structure of network and observed set of steady states, and (iv) sometimes we can predict unknown states or unknown regulations from an observed set of steady states alone. We illustrate the method by several applications to an experimentally determined regulatory network for biological functions.