Towards Modeling in Large-Scale Genetic and Metabolic Networks

Mathematical modeling of large-scale genetic and metabolic networks remains an open area of research with profound implications for understanding biochemical signaling systems in both prokaryotic and eukaryotic organisms. In this work, we extend the biochemical modeling framework based on the Power Law formalism , originally developed by M.A. Savageau to elucidate design principles in small bacterial networks. This formalism captures gene regulation and metabolic fluxes through two fundamental sets of parameters: kinetic parameters and kinetic orders . The kinetic parameters determine the magnitudes of fluxes, while the kinetic orders appearing as exponents in the power-law representation encode the regulatory structure of the system. We propose a mathematical technique to characterize the entire family of systems associated with the space of kinetic orders. We show that this exponent space possesses a well-defined structure in toric geometry and can therefore be modeled using algebraic varieties . To represent large-scale networks, we associate a toric variety with each polynomial equation of the system and then use geometric techniques to coherently combine these varieties into a global structure, or fan . This construction enables a modular and geometrically consistent representation of complex biochemical networks.