This Demonstration implements a mathematical model of the "repressilator" gene network. The repressilator consists of three genes, each of which represses the following gene. Because of the odd number of components in the circuit, there is a net negative feedback, which can give rise to oscillatory behavior (i.e. a limit cycle) for certain parameter values. The behavior can be visualized either as a function of time or in concentration phase space, where the three coordinates of each point represent the three repressor concentrations at a particular time.


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The repressilator was described and implemented in E. coli in [1]. A simplified version of Elowitz and Leibler's mathematical model is implemented here; we omit consideration of mRNA and focus exclusively on the protein (repressor) concentrations. The parameters of this implementation are: the ratio of the repressor production rate at a concentration of 1 to the repressor degradation rate, and the Hill function exponent , which specifies the cooperativity of the repressor activity. It is possible to show that sustained oscillations occur for values of approximately greater than . This implies that sustained oscillations can be achieved at small values of for large values of , but only at large values of for smaller values of . Interestingly, it is not possible to achieve sustained oscillations at all for , irrespective of the values of , which is borne out in the Demonstration.
[1] M. B. Elowitz and S. Leibler, "A Synthetic Oscillatory Network of Transcriptional Regulators," Nature, 403, 2000 pp. 335–338.
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