On the conversion of multivalued gene regulatory networks to Boolean dynamics

We consider the modeling approach introduced by R. Thomas for the qualitative study of gene regulatory networks. Tools and results on regulatory networks are often concerned only with the Boolean case of this formalism. However, multivalued approaches are sometimes more suited to model biological situations. Multivalued networks can be converted to partial Boolean maps, in a way that preserves the asynchronous dynamics. We ask whether this map can provide information on the original multivalued function, in particular via the application of Thomas’ rules. The problem of extending these partial Boolean maps to non-admissible states, i.e. states that do not have a multivalued counterpart, is also investigated. We observe that attractors are preserved if a “unitary” version of the original function is considered for conversion. Different extensions of the Boolean counterpart affect the structure of the regulatory graph in different ways. A particular technique for extending the Boolean unitary version of the network is identified, that ensures that no new circuits are added. This property, combined with the preservation of the asymptotic behaviour, can prove useful for the application of results and analyses defined in the Boolean setting to multivalued networks, and vice versa. By considering the conversion of a known example for the discrete multivalued case, we create a Boolean map showing that, for n ≥ 6, the absence of fixed points is compatible with the absence of local negative circuits in the regulatory graph.