Correspondence of Trap Spaces in Different Models of Bioregulatory Networks

Abstract

Mathematical models for bioregulatory networks can be based on different formalisms, depending on the quality of available data and the research question to be answered. Discrete Boolean models can be constructed based on qualitative data, which are frequently available. On the other hand, continuous models in terms of ordinary differential equations (ODEs) can incorporate time-series data and give more detailed insight into the dynamics of the underlying system. A few years ago, a method based on multivariate polynomial interpolation and Hill functions was developed for an automatic conversion of Boolean models to systems of ODEs. This method is frequently used by modelers in systems biology today, but there are only a few results available about the conservation of mathematical structures and properties across the formalisms. Here, we consider subsets of the phase space where some components stay fixed, called trap spaces, and demonstrate how Boolean trap spaces can be linked to invariant sets in the continuous state space. This knowledge is of practical relevance since finding trap spaces in the Boolean setting, which is relatively easy, allows for the construction of reduced ODE models.