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.
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.
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. 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