A mathematical framework for describing and analysing gene regulatory networks.

This paper presents a mathematical framework for describing and analysing gene regulatory networks by autonomous differential equations. It represents an improvement on existing frameworks in that it may handle a wider range of gene regulatory mechanisms. Gene regulatory networks are frequently threshold-dominated, i.e. genes are activated only when the concentration of certain gene products lie between definite thresholds. Here, the concept of regulatory domain is introduced to describe these regions in the phase space. To each regulatory domain is associated an indicator function whose value is 1 inside and 0 outside the domain. The indicator functions thus reflect the logical structure of the network. The sharp borders between the regulatory domains may be smoothed by replacing the logical step functions by continuous sigmoids or so-called logoid functions. A logoid function coincides with the step function outside a narrow interval around the threshold, and rises continuously from 0 to 1 inside it. Using logoids, the task of finding steady states is considerably simplified. A list of regions in phase space comprising all steady states lying close to a threshold is obtained by examining a certain type of matrix called the Logoid-Jacobian. In addition, this matrix leads to the conditions necessary for stability of the steady states. External signals may be conveniently incorporated in the form of Boolean variables. Thus the framework is well suited for studying gene regulatory networks both in single cells and multicellular systems.