Genetic-metabolic networks can be modeled as toric varieties

The mathematical modelling of genetic-metabolic networks is of outmost importance in the field of systems biology. Different formalisms and a huge variety of classical mathematical tools have been used to describe and analyse such networks. Michael A. Savageau defined a formalism to model genetic-metabolic networks called S-Systems, [16], [17], [19]. There is a limit in the number of nodes that can be analysed when these systems are solved using classical numerical methods such as non-linear dynamic analysis and linear optimization algorithms. We propose to use toric algebraic geometry to solve S-systems. In this work we prove that S-systems are toric varieties and that as a consequence Hilbert basis can be used to solve them. This is achieved by applying two theorems, proved here, the theorem about Embedding of S-Systems in toric varieties and the theorem about Toric Resolution on S-Systems. In addition, we define the realization of minimal phenotypic polytopes, phenotypic toric ideals and phenotypic toric varieties as a generalization of the phenotypic polytopes described by M. Savageau, [12], [13], [14]. The implications of the results here presented is that, in principle, they will facilitate solving large scale genetic-metabolic networks by means of toric algebraic geometry tools as well as facilitate elucidating their dynamics.