Using monoids for the integration of genomic and metabolic parameters in the prediction of phenotypes in regulatory cascades

Molecular network modeling requires the use of mathematical and computational formalisms for a robust and accurate prediction of phenotypes. Furthermore, there is a need to extend these formalisms to be applied to large-scale molecular networks, thus helping in the understanding of biological complexity. In this work, we propose an extension of the modeling framework known as design principles, developed by M.A. Savageau, which is based on power-law modeling. While power-law modeling has proven valuable for understanding various properties of molecular networks, it does not allow for the inference of kinetic orders, which are typically associated with the number of binding sites present in molecules. We address this limitation by modifying the traditional approach to incorporate the use of monoids, thereby introducing a novel methodological framework we call Genotype Arithmetic. The resulting combinatorial object defines a family of geometric points, whose fixed points in the exponent space impose a set of constraints that determine the appropriate kinetic order to be used in the power law models. This feature enables a prediction of the binding strength and/or the number of DNA binding sites in regulatory sequences, as well as the reaction orders in enzymatic kinetics. To demonstrate the applicability of the present approach, we illustrated how the number of binding sites can be approximated in metabolic pathways formed by 3 to 9 reactions, in allosteric systems of end-product inhibition with intermediate catalytic reactions and one gene inhibition.