Abstracts Biological Sciences

Mathematical Models of Biochemical Oscillations

by Emery David Conrad

The goal of this paper is to explain the mathematics involved in modeling biochemical oscillations. We first discuss several important biochemical concepts fundamental to the construction of descriptive mathematical models. We review the basic theory of differential equations and stability analysis as it relates to two-variable models exhibiting oscillatory behavior. The importance of the Hopf Bifurcation will be discussed in detail for the central role it plays in limit cycle behavior and instability. Once we have exposed the necessary mathematical framework, we consider several specific models of biochemical oscillators in three or more variables. This will include a detailed analysis of Goodwin's equations and their modification first studied by Painter. Additionally, we consider the consequences of introducing both distributed and discrete time delay into Goodwin's model. We will show that the presence of distributed time lag modifies Goodwin's model in no significant way. The final section of the paper will discuss discrete time lag in the context of a minimal model of the circadian rhythm. In the main, this paper will address mathematical, as opposed to biochemical, issues. Nevertheless, the significance of the mathematics to the biochemistry will be considered throughout.