Stability analysis of an autocatalytic protein model

A self-regulatory genetic circuit, where a protein acts as a positive regulator of its own production, is known to be the simplest biological network with a positive feedback loop. Although at least three components—DNA, RNA, and the protein—are required to form such a circuit, stability analysis of the fixed points of this self-regulatory circuit has been performed only after reducing the system to a two-component system, either by assuming a fast equilibration of the DNA component or by removing the RNA component. Here, stability of the fixed points of the three-component positive feedback loop is analyzed by obtaining eigenvalues of the full three-dimensional Hessian matrix. In addition to rigorously identifying the stable fixed points and saddle points, detailed information about the system can be obtained, such as the existence of complex eigenvalues near a fixed point.