Positive quadratic system representation of molecular interaction in a cell

This paper proposes a new system model called a positive quadratic system for molecular interaction in a cell, including a signal transduction pathway and a gene regulatory pathway. Although the whole network of these pathways is far more complex than simple chemical reactions, one of the features of the proposed model is that it can capture the quadratic property of molecular interaction as well as the positivity properties of the state variables with a rather simple form. Based on this model, we derive a sufficient condition of the stability in the sense of Lyapunov, and present a method for estimating a positive invariant set depending on the initial state. We show that the proposed method is effective by numerical simulations of two fundamental biological models.