Modularity plays a fundamental role in the prediction of the behavior of a system from the behavior of its components, guaranteeing that the properties of individual components do not change upon interconnection. Just as electrical, hydraulic, and other physical systems often do not display modularity, nor do many biochemical systems, and specifically, genetic networks. Here, we study the effect of interconnections on the input–output dynamic characteristics of transcriptional components, focusing on a property, which we call ‘retroactivity’, that plays a role analogous to non-zero output impedance in electrical systems. In transcriptional networks, retroactivity is large when the amount of transcription factor is comparable to, or smaller than, the amount of promoter-binding sites, or when the affinity of such binding sites is high. To attenuate the effect of retroactivity, we propose a feedback mechanism inspired by the design of amplifiers in electronics. We introduce, in particular, a mechanism based on a phosphorylation–dephosphorylation cycle. This mechanism enjoys a remarkable insulation property, due to the fast timescales of the phosphorylation and dephosphorylation reactions.
In engineering, modularity of components is a key property allowing the development of complex networks with advanced functionalities. Modularity guarantees that the input–output behavior of a component does not change upon interconnection. Here, we explore conditions affecting modularity of biological genetic circuitry, and explore several mechanisms for increasing modularity. As expected from engineering studies of electrical, mechanical, and hydraulic systems, the property of modularity does not generally hold in biological systems. We refer to retroactivity as the phenomenon by which the behavior of an upstream component is affected by the connection to a downstream component. For general interconnections between genetic regulatory modules, the retroactivity is non-zero and can have a dramatic effect on system behavior. We first analyze the dynamics of a genetic regulatory component (Figure 2) in isolation, and then we quantify the change in its dynamics, the retroactivity, due to the interconnection with other genetic modules (Figure 3). Retroactivity is large when the amount of transcription factor is comparable to or smaller than the amount of promoter-binding sites or when the affinity of such binding sites is high. We next show how insulation between an upstream component and a downstream one can be attained by connecting them through an insulation device. An insulation device is a component that (1) maintains the same output independently of the number of downstream clients that are fed by the output and (2) does not affect the upstream component from which it receives the signal. The general insulation mechanism that we propose is inspired by the design of non-inverting amplifiers in electronics. It relies on a large input amplification gain and on a similarly large negative output feedback. Two biological realizations of this general mechanism are analyzed in detail. The first one involves a strong, non-leaky promoter to implement a large input gain, combined with an abundant protease that degrades the protein product and hence implements high gain feedback. The second one involves a post-translational modification mechanism through a phosphorylation–dephosphorylation cycle such as found in MAPK cascades. Our dynamic analysis reveals that a simple phosphorylation–dephosphorylation cycle enjoys a remarkable insulation property. Such a property is in part due to the fast timescales of phosphorylation–dephosphorylation reactions. Such a mechanism, as a signal transduction system, has thus an inherent capacity to provide insulation and hence to increase the modularity of the system in which it is placed.