The ability to adapt to stimuli is a defining feature of many biological systems and critical to maintaining homeostasis. While it is well appreciated that negative feedback can be used to achieve homeostasis when networks behave deterministically, the effect of noise on their regulatory function is not understood. Here, we combine probability and control theory to develop a theory of biological regulation that explicitly takes into account the noisy nature of biochemical reactions. We introduce tools for the analysis and design of robust homeostatic circuits and propose a new regulation motif, which we call antithetic integral feedback. This motif exploits stochastic noise, allowing it to achieve precise regulation in scenarios where similar deterministic regulation fails. Specifically, antithetic integral feedback preserves the stability of the overall network, steers the population of any regulated species to a desired set point, and adapts perfectly. We suggest that this motif may be prevalent in endogenous biological circuits and useful when creating synthetic circuits.

Homeostasis is a recurring theme in biology. Homeostatic mechanisms commonly ensure that regulated variables robustly and completely adapt to environmental perturbations. This robust perfect adaptation (RPA) feature is achieved by incorporating mathematical integration in a negative feedback strategy. 1, 2 Despite its benefits in natural circuits, the synthetic realization of integral feedback has remained elusive due to the complexity of the required biological computations. Here we first mathematically prove that there is fundamentally a single biomolecular controller topology 3 that realizes integral feedback for arbitrary intracellular networks with noisy dynamics. Such a topology guarantees RPA for both the cell populationaverage and for the time-average of single cells. We then develop the first synthetic gene network implementation of such an integral controller in a living cell, 4 and demonstrate its tunability and adaptation properties. A growth control application shows the inherent capacity of our integral feedback controller to deliver robustness, and highlights its potential use as a versatile controller for regulation of biological variables in uncertain networks. Our results provide new conceptual and practical tools in the area of Cybergenetics 3,5 where control theory and synthetic biology come together to enable the engineering of novel synthetic controllers that steer the dynamics of living systems. [3][4][5][6][7][8][9] Integral feedback control is arguably one the most fundamental regulation strategies in engineering practice. From modern jetliners to industrial plants, integral feedback loops reliably drive physical variables to their desired values with great robustness and precision. 10 It is becoming increasingly appreciated that

As a result of an author oversight in the originally published version of this article, the word ''Biomolecular'' was misspelled as ''Bimolecular'' in the title. This error has now been corrected in the article online. The authors apologize for the error.

An efficient continuum model for simulating polymer blends and copolymers is presented. In this model, the interactions are short-range and purely repulsive, thus allowing for excellent computational performances. The driving force for phase separation is a difference in the repulsive interaction strength between like and unlike mers. The model consists essentially of a molecular-dynamics algorithm, supplemented by an appropriate Monte Carlo exchange process. To demonstrate the effectiveness of the model we study two systems, a symmetric binary blend of polymers and a symmetric diblock copolymer system. For the binary blend, we determine the phase diagram and find, as predicted by theory, that the critical interaction parameter scales with the inverse of the chain length of the polymers. For the diblock copolymer system, we study both the one-phase region and the microphase separated lamellar region. For the latter, we show that constant-pressure algorithms are more appropriate since, contrary to recent lattice simulations, the lamellar spacing can self-adjust in such an ensemble.

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