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Synthetic biology opens up the possibility of creating circuits that would not survive in the natural world and studying their behaviors in living cells, expanding our notion of biology. Based on this, we analyze on a synthetic biological system the effect of coupling between two instability-generating mechanisms. The systems considered are two topologically equivalent synthetic networks. In addition to simple periodic oscillations and stable steady state, the system can exhibit a variety of new modes of dynamic behavior: coexistence between two stable periodic regimes (birhythmicity) and coexistence of a stable periodic regime with a stable steady state (hard excitation). Birhythmicity and hard excitation have been proved to exist in biochemical networks. Through bifurcation analysis on these two synthetic cellular networks, we analyze the function of network structure for the collapse and revival of birhythmicity and hard excitation with the variation of parameters. The results have illustrated that the bifurcation space can be divided into four subspaces for which the dynamical behaviors of the system are generically distinct. Our analysis corroborates the results obtained by numerical simulation of the dynamics.

The successful construction of the first synthetic oscillator in 2000 signaled the entry into the new era of artificial cellular rhythms. Known as the repressilator, this oscillator, expressed in

As a result of interactions between numerous intracellular or extracellular biomolecules, complex cellular behaviors could be easily found, such as bistability [

Birhythmicity, namely, the coexistence between two stable regimes of limit cycle oscillations, had been reported in the various types of complex

Moreover, it is ubiquitous for coupled feedback loops occurring in many contexts (such as metabolism, signalling, and development) to control important aspects of cell physiology, for example, circadian rhythms [

However, such feedback loops are usually found as a coupled structure rather than a single isolated form in various cellular circuits [

Schematic diagrams of the coupled two negative feedback genetic circuits. Panel (a) is the transcriptional regulation networks for two different coupled negative feedback loops. panel (b) is the simplified and separated schematic diagram of panel (a). In panel (b), the black circle and the white circle are of opposite regulating functions, namely, “activation

The central negative feedback loop consisting of three repressors (

Based on our modeling method, we gave the mathematical equation about the network shown in Figure

Since it is common for cellular networks composed of complicated interconnections among components, those subnetworks of particular functioning are often identified as network motifs. Intriguingly, among such network motifs, feedback loops are very often found as a coupled structure in cellular circuits, which are thought of playing important dynamical roles. Among these synthetic integrated genetic circuits, we mainly focus on one negative feedback loop with three repressors plus one additional feedback loop by adding another regulator. The coupled feedback loops with different topology structures are shown in Figure

In order to investigate the potential dynamical behaviors of cellular circuits, which are in general quite complicated due to the nonlinear interaction among the components, we model the interconnected negative feedback loops in the present transcriptional regulatory network with Hill’s kinetics without considering the kinetic equation of genes. When component

When the regulations among components

Otherwise, when the regulations among these three components are “repression

In the above two mathematical models,

By using XPPAUT, we identify the ranges of the parameter,

The mathematic equations for networks in Figures

From Figure

Bifurcation diagram of component

According to the investigation above, there are four bifurcation points: two limit points 14.91 and 21.32; two Hopf bifurcation points 15.85 and 20.85. As

What is more, in Figure

Typical time courses of state variable

Models of synthetic genetic applets usually either consist of single synthetic units [

The further bifurcation analysis about two parameters is performed using the XPPAUT and Matlab packages for two topological equivalent coupled genetic oscillators and shows that already two negative feedback loops provide a large variety of possible regimes. Figures

Bifurcation diagrams of two parameters showing the domains of different dynamical behaviors for model (

Bifurcation diagram of parameter

Bifurcation diagram of parameter

Bifurcation diagram of parameter

Bifurcation diagrams of two parameters showing the domains of different dynamical behaviors for model (

Bifurcation diagram of parameter

Bifurcation diagram of parameter

Bifurcation diagram of parameter

From Figure

Figure

Figure

Figure

With the same analysis as above about the network shown in Figure

In model (

From Figure

In Figure

From Figure

Feedback loops are omnipresent in natural cellular circuits. There have been a lot of functions reported for single feedback loops in recent years; that is, negative feedback loops could reduce response signal amplitude and response time, maintain homeostasis, and are sufficient for oscillation, especially the recent research synthesized oscillator, namely, repressilator, in

In this paper, we have investigated the dynamics behaviors of coupled feedback loops. The coupled feedback loops considered here are composed of one central negative feedback loop pluse another additional feedback loops. There could be twelve topology structures for the coupled feedback loops that are added by only one node and two regulations. Based on the components’ number of the additional feedback loops, we divide the twelve systems into three types. With the help of XPPAUT and Matlab, we obtained the bifurcation diagrams for every system. Through numerical analysis, we find that the complex oscillatory phenomena including birhythmicity and hard excitation exist in both of these two models. By comparative analysis of two parameters bifurcation, we proved that there are almost the same bifurcation phenomena in models (

The present results on the occurrence of complex oscillatory phenomena in our synthetic model are of particular interest for understanding the conditions in which birhythmicity may arise in biological systems, such as

Birhythmicity requires stringent conditions both on the kinetics and on the parameter values. Thus, it is probably less frequent than its well-known stationary counterpart, bistability, in which two stable steady states coexist for a given set of experimental conditions, as demonstrated for several biochemical systems such as the peroxidase reaction. Birhythmicity provides a new mode of physiological regulation as it allows for a switch between two periodic regimes upon suitable perturbation. It would be of interest to search for this phenomenon not only in chemical or metabolic oscillatory systems but also in the many rhythmic processes occurring in the brain, which arise precisely from multiple regulatory interactions between neurons.

Although the relative smallness of these domains raises doubts about the possible physiological significance of birhythmicity in regard to circadian rhythm generation, beyond the particular context of circadian rhythms, we discuss the results in the light of other mechanisms underlying birhythmicity and inhomogeneous limit cycles in regulated biological systems. Furthermore, the development of artificial cellular oscillators opens the way to pharmacological applications such as pulsatile drug delivery.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors acknowledge the support from the Natural Science Foundation (nos. 11105040 and 61104138) of China, the Natural Science Foundation of the Education Department of Henan, China (no. 2011B110003), and the Excellent Young Scientific Talents Cultivation Foundation of Henan University (no. 0000A40351).

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