A MATHEMATICAL FRAMEWORK FOR DESCRIBING AND ANALYZING GENE REGULATORYNETWORKS

This paper presents a mathematical framework for describing and analys ing 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 whe n the concentration of certain gene products lie between definite thre sholds. Here, the concept of regulatory domain is introduced to descri be these regions in the phase space. To each regulatory domain is asso ciated 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 sigmoid s or so-called logoid functions. A logoid function coincides with the step function outside a narrow interval around the threshold, and rise s continuously from 0 to I inside it. Using logoids, the task of findi ng steady states is considerably simplified. A list of regions in phas e space comprising all steady states lying close to a threshold is obt ained by examining a certain type of matrix called the Logoid-Jacobian . In addition, this matrix leads to the conditions necessary for stabi lity of the steady states. External signals may be conveniently incorp orated in the form of Boolean variables. Thus the framework is well su ited for studying gene regulatory networks both in single cells and mu lticellular systems. (C) 1995 Academic Press Limited