Robust tunable transcriptional oscillators using dynamic inversion

We present a theory and associated algorithms to synthesize controllers that may be used to build robust tunable oscillations in biological networks. As an illustration, we build robust tunable oscillations in the celebrated repressilator synthesized by Elowitz and Leibler. The desired oscillations in a set of mRNA's and proteins are obtained by injecting an oscillatory input as a reference and by synthesizing a dynamic inversion based tracking controller. This approach ensures that the repressilator can exhibit oscillations irrespective of (1) the maximum number of proteins per cell and (2) the ratio of the protein lifetimes to the mRNA lifetimes. The frequency and the amplitude of at least one output (either mRNA or protein) can now be controlled arbitrarily. In addition, we characterize the gain stability of this 3-node network and generalize it to the case of -node networks.