Topologies of Phosphorylation Cycles As Insulation Devices

One of the problems faced in building large circuits from small parts or modules in synthetic biology is that of retroactivity. Retroactivity is the change in the dynamics of a system when it is connected with another system or load. An insulation device is a system that can be connected in between such modules to attenuate the effect of retroactivity, enabling the design of larger circuits. A single phosphorylation cycle has been theoretically and experimentally shown to have the ability to attenuate the effect of retroactivity applied by downstream systems on genetic circuits due to its fast reaction rates. However, it has been found that this retroactivity attenuation property comes at the expense of an increased retroactivity to the input, wherein the cycle slows down the signal it receives from the upstream system. In this work, we provide a library of insulation devices with different insulation properties, built from various topologies of phosphorylation cycles, utilizing its fast timescale of operation. We find that cascades of phosphorylation cycles break the tradeoff faced by a single cycle, allowing the attenuation of retroactivity applied by downstream systems, while applying a small retroactivity to the input. We show that there is an optimal number of cycles that maximally extends the linear operating region of the insulation device while keeping the desired retroactivity attenuation properties, when a common phosphatase is used. Double phosphorylation cycles can show similar retroactivity attenuation properties to a single cycle, and provide either an ultrasensitive response to the input, or a linear response with a sign-sensitive delay. These findings provide insights into cellular signaling systems, where such phosphorylation cycles are ubiquitous. In addition, they provide a variety of insulation device designs for synthetic biology applications.