On the dynamics of a simple biochemical control circuit

The quantitative dynamics of a biochemical control circuit that regulates enzyme or protein synthesis by end-product feedback is analyzed. We first study a simplified repressible system, which is known to exhibit either a steady state or an oscillatory solution. By showing the analogy of this n-dimensional system with a time-delay equation for a single variable the mechanism of the self-sustained oscillations becomes transparent. In a more sophisticated system we will find as well either steady state or oscillatory solutions. We determine the role of the parameters with respect to stability and frequency. The most general case will be treated by means of the concept of Lyapunov exponents.