Some Basic Biological Motifs And Their Stability

Due to the complexity of the biological system, the nonlinear dynamical tools, including stability analysis, feedback control theory, bifurcation analysis, are needed in understanding the mechanism of biological regulatory network. On the contrary, the deep study of biological system brings new developments of nonlinear dynamics.There is a wide variety of regulation ways inside of cells, including transcriptional regulation, post-translational regulation, metabolic regulation, Electrophysiological regulation, etc. Every regulation way has its distinct biological property, and therefore has its unique mathematical model.The main task of this paper is to study the basic motifs simplified from the original biological regulatory network. The two-dimensional biological motifs were transformed into two-dimensional differential equations through the mathematical modeling. The fundamental mechanism behind the biochemical phenomenon is exposited by analyzing the characteristic matrix and the eigenvalues of the two-dimensional differential equation, and using nonlinear dynamics tools for the theoretical studies. In this paper, the two-dimensional system with a feedback loop, which contained positive feedback loop of A and B mutually activating each other, positive feedback loop of A and B mutually inhibiting each other and negative feedback loop of A promoting B, B inhibiting A, was investigated. In addition, the feedback loop of A and B mutually transforming was also discussed in the paper.Through theoretical analysis, we have the following three major findings:(1) All the twelve cases of the positive feedback loop of A and B mutually activating each other (without transforming between A and B) are unstable under certain parameter sets;(2) all the twelve cases of positive feedback loop of A and B mutually inhibiting each other (without transforming between A and B) are unstable under certain parameter sets;(3) all the twelve cases of the negative feedback loop of A promoting B, B inhibiting A are stable under all parameter sets;(4) The feedback loop of A and B mutually conversing was partially worked out. Thus the following results are obtained.1) The system could be unstable when positive feedback loop exists in the two-dimensional system (the case of A and B mutually transforming isn’t included).2) All the steady-state solutions of the system are stable when only one negative feedback loop appears in the two-dimensional system (the case of A and B mutually transforming isn’t included).3) The systems containing transforming are more complicated.The last finding hasn’t been fully studied due to lack of time, and the uncompleted part will be conducted in the future work.