Effective parameters determining the information flow in hierarchical biological systems.

Signaling networks are abundant in higher organisms. They play pivotal roles, e.g., during embryonic development or within the immune system. In this contribution, we study the combined effect of the various kinetic parameters on the dynamics of signal transduction. To this end, we consider hierarchical complex systems as prototypes of signaling networks. For given topology, the output of these networks is determined by an interplay of the single parameters. For different kinetics, we describe this by algebraic expressions, the so-called effective parameters.When modeling switch-like interactions by Heaviside step functions, we obtain these effective parameters recursively from the interaction graph. They can be visualized as directed trees, which allows us to easily determine the global effect of single kinetic parameters on the system's behavior. We provide evidence that these results generalize to sigmoidal Hill kinetics.In the case of linear activation functions, we again show that the algebraic expressions can be immediately inferred from the topology of the interaction network. This allows us to transform time-consuming analytic solutions of differential equations into a simple graph-theoretic problem. In this context, we also discuss the impact of our work on parameter estimation problems. An issue is that even the fitting of identifiable effective parameters often turns out to be numerically ill-conditioned. We demonstrate that this fitting problem can be reformulated as the problem of fitting exponential sums, for which robust algorithms exist.