Representing Model Ensembles as Boolean Functions

Abstract

Families of ODE models ẋ = f (x) characterized by a common sign structure Σ of their Jacobi matrix J(f ) are investigated within the formalism of qualitative differential equations [4]. In the context of regulatory networks the sign structure of the Jacobi matrix carries the information about which components of the network inhibit or activate each other. Information about constraints on the behavior of models in this family is stored in a so called qualitative state transition graph G QDE (Σ) [4]. We showed previously that a similar approach can be used to analyze a model pool of Boolean functions characterized by a common interaction graph [13]. Here we show that the opposite approach is fruitful as well. We show that the qualitative state transition graph G QDE (Σ) can be reduced to a "skeleton" represented by a Boolean function f Σ conserving the reachability properties. This reduction has the advantage that approaches such as model checking and network inference methods can be applied to G async (f Σ ) within the framework of Boolean networks. Furthermore, our work constitutes an alternative to approaches like [14] and [15] to link Boolean networks and differential equations.