Global Entrainment of Transcriptional Systems to Periodic Inputs

This paper addresses the problem of providing mathematical conditions that allow one to ensure that biological networks, such as transcriptional systems, can be globally entrained to external periodic inputs. Despite appearing obvious at first, this is by no means a generic property of nonlinear dynamical systems. Through the use of contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all their solutions converge to a fixed limit cycle. General results are proved, and the properties are verified in the specific cases of models of transcriptional systems as well as constructs of interest in synthetic biology. A self-contained exposition of all needed results is given in the paper.Author Summary: The activities of living organisms are governed by complex sets of biochemical reactions. Often, entrainment to certain external signals helps control the timing and sequencing of reactions. An important open problem is to understand the onset of entrainment and under what conditions it can be ensured in the presence of uncertainties, noise, and environmental variations. In this paper, we focus mainly on transcriptional systems, modeled by Ordinary Differential Equations. These are basic building blocks for more complex biochemical systems. However, the results that we obtain are of more generality. To illustrate this generality, and to emphasize the use of our techniques in synthetic biology, we discuss the entrainment of a Repressilator circuit and the synchronization of a network of Repressilators. We answer the following two questions: 1) What are the dynamical mechanisms that ensure the entrainment to periodic inputs in transcriptional modules? 2) Starting from natural systems, what properties can be used to design novel synthetic biological circuits that can be entrained? For some biological systems which are always “in contact” with a continuously changing environment, entrainment may be a “desired” property. Thus, answering the above two questions is of fundamental importance. While entrainment may appear obvious at first thought, it is not a generic property of nonlinear dynamical systems. The main result of our paper shows that, even if the transcriptional modules are modeled by nonlinear ODEs, they can be entrained by any (positive) periodic signal. Surprisingly, such a property is preserved if the system parameters are varied: entrainment is obtained independently of the particular biochemical conditions. We prove that combinations of the above transcriptional module also show the same property. Finally, we show how the developed tools can be applied to design synthetic biochemical systems guaranteed to exhibit entrainment.