Modeling the tunability of the dual-feedback genetic oscillator.

Oscillatory gene circuits are ubiquitous to biology and are involved in fundamental processes of cell cycle, circadian rhythms, and developmental systems. The synthesis of small, non-natural oscillatory genetic circuits has been increasingly used to test the fundamental principles of genetic network dynamics. While the "repressilator" was used to first demonstrate the proof of principle, a more recently developed dual-feedback, fast, tunable genetic oscillator has demonstrated a greater degree of robustness and control over oscillatory behavior by combining positive- and negative-feedback loops. This oscillator, combining lacI (negative-) and araC (positive-) feedback loops, was, however, modeled using multiple layers of differential equations to capture the molecular complexity of regulation, in order to explain the experimentally measured oscillations. In the search for design principles of such minimal oscillatory circuits, we have developed a reduced model of this dual-feedback loop oscillator consisting of just six differential equations, two of which are delay differential equations. The delay term is optimized, as the only free parameter, to fit the experimental dynamics of the oscillator period and amplitude tunability by the two inducers isopropyl β-D-1-thiogalactopyranoside (IPTG) and arabinose. We proceed to use our reduced and experimentally validated model to redesign the network by comparing the effect of asymmetry in gene expression at the level of (a) DNA copy numbers and the rates of (b) mRNA translation and (c) degradation, since experimental and theoretical work had predicted a need for an asymmetry in the copy numbers of activator (araC) and repressor (lacI) genes encoded on plasmids. We confirm that the minimal period of the oscillator is sensitive to DNA copy number asymmetry, and can demonstrate that while the asymmetry in the translation rate has an identical effect as the plasmid copy numbers, modulating the asymmetry in mRNA degradation can improve the tunability of the period and amplitude of the oscillator. Thus, our model predicts control at the level of translation can be used to redesign such networks, for improved tunability, while at the same time making the network robust to replication "noise" and the effects of the host cell cycle. Thus, our model predicts experimentally testable principles to redesign a potentially more robust oscillatory genetic network.