Modular cell biology: retroactivity and insulation
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Domitilla Del Vecchio1, Alexander J Ninfa2 & Eduardo D Sontag3
- Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA
- Department of Biological Chemistry, University of Michigan Medical School, Ann Arbor, MI, USA
- Department of Mathematics, Rutgers, The State University of New Jersey, Piscataway, NJ, USA
Correspondence to: Domitilla Del Vecchio1 Department of Electrical Engineering and Computer Science, University of Michigan, 1301 Beal Avenue, Ann Arbor, MI 48109, USA. Tel.: +1-734-764-6581; Fax: +1-734-763-8041; Email: ddv@umich.edu
Received 28 September 2007; Accepted 30 November 2007; Published online 12 February 2008
Article highlights
- The general concept of retroactivity has been introduced to model any change in the dynamic behavior of an upstream system due to the interconnection with a downstream system, thus formally extending the notion of non-zero output impedance to non-electrical systems.
- A scalable procedure for operationally quantifying the effects of the retroactivity on the dynamics of an upstream system has been proposed, which applies to transcriptional and signal transduction networks.
- A general mechanism relying on large amplification and large negative feedback has been mathematically and computationally shown to attain attenuation of the retroactivity effects on transcriptional and signal transduction modules. Accordingly, insulation device designs have been proposed.
Synopsis
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.
Figure 2
The transcriptional component takes as input u protein concentration Z and gives as output y protein concentration X. The transcription factor Z binds to operator sites on the promoter. The red part belongs to a downstream transcriptional block that takes protein concentration X as its input.
Full figure and legend (81K)Figures & Tables indexFigure 3
Simulation results for the system in equation (3). Here, k(t)=0.01(1+sin(
t)) with =0.005, kon=10, koff=10, 
=0.01, pTOT=100, X(0)=5. The choice of protein decay rate (in min-1) corresponds to a half-life of about 1 h. The frequency of oscillations is chosen to have a period of about 12 times the protein half-life in accordance to what is experimentally observed in the synthetic clock of Atkinson et al (2003). All simulation results were obtained by using MATLAB (Simulink), with variable-step ODE solver ODE23s. The green plot (solid line) represents X(t) originating by the isolated system in equation (2), while the blue plot (dashed line) represents X(t) obtained by the interconnected system of equation (3). Both transient and permanent behaviors are different.
Acknowledgements
We thank Peter Woolf and Richard M. Murray for discussions that were very relevant to the development of this work. We also thank the reviewers for providing additional references and for their suggestions that improved the content and the readability of the paper. This work was supported in part by NSF grant DMS-0614371, in part by grant R01GM063642 from the NIGMS, and in part by the Rackham Faculty Research Grant from University of Michigan.
