Editing Brian Goodwin (section)

==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

:<math>
\frac{dX}{dt}= {k_1 \over K_1 + Z^n}-k_2X
</math>

:<math>
\frac{dY}{dt}= k_3 X - k_4 Y
</math>

:<math>
\frac{dZ}{dt}= k_5 Y - k_6 Z
</math>

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 rhythm|circadian clocks]], [[cell division]] or physiological control systems.