Gene Networks and Development
Shortly after François Jacob and Jacques Monod developed their first model of gene regulation, Goodwin proposed the first model of a genetic oscillator, showing that regulatory interactions among genes allowed periodic fluctuations to occur. Shortly after this model became published, he also formulated a general theory of complex gene regulatory networks using statistical mechanics. In its simplest form, Goodwin's oscillator involves a single gene that represses itself. Goodwin equations were originally formulated in terms of conservative (Hamiltonian) systems, thus not taking into account dissipative effects that are required in a realistic approach to regulatory phenomena in biology. Many versions have been developed since then. The simplest (but realistic) formulation considers three variables, X, Y and Z indicating the concentrations of RNA, protein and end product which generates the negative feedback loop.The equations are
and closed oscillations can occur for n>8 and behave limit cycles: after a perturbation of the system's state, it returns to its previous attractor. A simple modification of this model, adding other terms introducing additional steps in the transcription machinery allows to find oscillations for smaller n values. Goodwin's model and its extensions have been widely used over the years as the basic skeleton for other models of oscillatory behavior, including circadian clocks, cell division or physiological control systems.
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