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Nonequilibrium stochastic reaction networks are commonly found in both biological and nonbiological systems, but have remained hard to analyze because small differences in rate functions or topology can change the dynamics drastically. Here, we conjecture exact quantitative inequalities that relate the extent of fluctuations in connected components, for various network topologies. Specifically, we find that regardless of how two components affect each other's production rates, it is impossible to suppress fluctuations below the uncontrolled equivalents for both components: one must increase its fluctuations for the other to be suppressed. For systems in which components control each other in ringlike structures, it appears that fluctuations can only be suppressed in one component if all other components instead increase fluctuations, compared to the case without control. Even the general $N$-component system---with arbitrary connections and parameters---must have at least one component with increased fluctuations to reduce fluctuations in others. In connected reaction networks it thus appears impossible to reduce the statistical uncertainty in all components, regardless of the control mechanisms or energy dissipation.

The first synthetic genetic oscillator or ‘repressilator’ is simplified using insights from stochastic theory, thus achieving remarkably precise and robust oscillations and informing current debates about the next generation of synthetic circuits and their potential applications in cell-based therapies. The first genetic oscillator, known as the 'repressilator', was arguably a leader in the field of synthetic biology at the beginning of this century. It is, however, notoriously unreliable. Now Johan Paulsson and colleagues have succeeded in reducing error propagation and information losses in the original design, not by adding control loops, but by removing existing features, achieving remarkably precise and robust oscillations. The fact that such precise behaviour can be achieved in a relatively simple circuit, despite interaction with a noisy organism, should instruct current debates about cell-based therapies and the reliability of synthetic approaches in complex and variable environments. Synthetically engineered genetic circuits can perform a wide variety of tasks but are generally less accurate than natural systems. Here we revisit the first synthetic genetic oscillator, the repressilator1, and modify it using principles from stochastic chemistry in single cells. Specifically, we sought to reduce error propagation and information losses, not by adding control loops, but by simply removing existing features. We show that this modification created highly regular and robust oscillations. Furthermore, some streamlined circuits kept 14 generation periods over a range of growth conditions and kept phase for hundreds of generations in single cells, allowing cells in flasks and colonies to oscillate synchronously without any coupling between them. Our results suggest that even the simplest synthetic genetic networks can achieve a precision that rivals natural systems, and emphasize the importance of noise analyses for circuit design in synthetic biology.

Nullclines provide a convenient way of characterising and understanding the behaviour of low dimensional nonlinear deterministic systems, but are, perhaps not unsurprisingly, a poor predictor of the behaviour of discrete state stochastic systems in the low numbers regime. Such models are appropriate in many biological systems. In this paper we propose a graphical discrete ‘nullcline-like’ construction, inspired by the Markov chain tree theorem, and investigate its application to the original genetic toggle switch, which is a feedback interconnection of two mutually repressing genes. When the feedback gain (the ‘cooperativity’) is sufficiently large, the deterministic system exhibits bistability, which shows itself as a bimodal stationary distribution in the discrete stochastic system for sufficiently large numbers. However, at small numbers a third mode appears corresponding to roughly equal numbers of each molecule. Without cooperativity, on the other hand (i.e. low feedback gain), the deterministic system has just one stable equilibrium. Nevertheless, the stochastic system can still exhibit bimodality. In this paper, we illustrate that the discrete ‘nullclines’ proposed can, without the need to calculate the steady state distribution, provide an efficient graphical way of predicting the shape of the stationary probability distribution in different parameter regimes, thus allowing for greater insights in the observed behaviours.