Multistability and Its Robustness of a Class of Biological Systems

Multistability of biological systems with complex nonlinear regulatory schemes is an important research topic in system biology. In many models of biological systems, the regulatory functions are of saturation type. The linear sectors, in which the saturation type functions reside, have been extensively adopted to deal with these saturation type functions. The stability analysis resulting from linear sectors is however often conservative as a wide linear section is required to include a large portion of a saturation type function. In this paper, we utilize piecewise linear sectors, recently adopted in nonlinear control theory, to investigate multistability of a class of biological systems with sum regulatory schemes. We will estimate the domain of attraction of each stable equilibrium and examine the robust stability of each equilibrium in the face of disturbances that are bounded in magnitude or energy. A genetic toggle switch in Escherichia coli is employed as an example to illustrate the applicability and effectiveness of our analysis method.