  The mathematical modelling of genetic-metabolic networks is of out most
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, [12], [13], [15]. 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, [8], [9], [10]. Also we studied
a synthetic oscillatory network of two genes under diverse environmental
control modes, we reproduce S. Savageau results and additionally we obtain an
invariant sustained oscillatory region. 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.
