Jacob and Monod proposed the genetic control of protein synthesis in the late 1950s. Goodwin, then Griffith, formulated models based on this theory using basic biochemical kinetics and several quasi-steady state assumptions. Create one or more mathematical models using biochemical kinetics, then converting the biochemical equations into differential equations that can be analyzed. You may want to detail a specific system, such as repression of the tryptophan gene or induction of the lac operon, stressing the biological/biochemical aspects of the model, or you may want to concentrate on the mathematical analysis, including bifurcations, such as Hopf bifurcations for the repression model or a saddle node bifurcation for induction.

References:

1. B. C. Goodwin: Oscillatory behavior in enyzmatic control processes, Adv. Enzyme Reg. 3 (1965), 425-439.
2. B. C. Goodwin: Temporal Organization in Cells, Academic Press, New York (1963).
3. J. S. Griffith: Mathematics of cellular control processes, I; II, J. Theor. Biol.20 (1968), 202-208, 209-216.
4. F. Jacob and J. Monod: On regulation of gene activity, Cold Spring Harbor Symp. Quant. Biol. 26 (1961), 389-401.
5. J. J. Tyson and H. G. Othmer: The dynamics of feedback control circuits in biochemical pathways, Prog. in Theor. Biol. (R. Rosen and F. M. Snell, eds.) Academic Press, New York (1978).
6. M. Santillan and M. C. Mackey: Dynamic regulation of the tryptophan operon: A modeling study and comparison with experimental data, Preprint (2000).