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Realizing “integral control” in living cells: How to overcome leaky integration due to dilution?

Feedback control systems with integral action have the unique ability to perfectly reach constant set-points and to reject constant disturbances. Due to these properties, they are ubiquitous in engineering systems and frequently found in natural biological systems that achieve homeostasis and adaptation. Recently, there has been increasing interest in realizing integral controllers in the synthetic biology community as a way to improve robustness of circuits and tightly regulate gene expression. Although a number of circuit designs have been proposed, their functionality often hinges on the critical assumption that species in the controller do not dilute, since dilution breaks the controller integration structure and leads to “leaky integration”. While this assumption is satisfied in cell-free systems, it is not met in implementations in living cells, where cell growth and division dictates species dilution. In this report, we abstract previously proposed designs of biomolecular integral controllers into two ideal integral control motifs (IICM), type I and type II. Based on these, we mathematically demonstrate how engineering a time-scale separation between the controller reactions and dilution, we can obtain quasi-integral control motifs (qICM) that recover almost perfect adaptation even in the presence of dilution. Contrary to previous hypothesis, our results show that implementing a fast integral action alone may not be sufficient to recover the adaptation property. Our results are based on a general dynamical model of a “plant” and a “controller” for which we provide easy-to-check algebraic conditions for almost perfect adaptation. These conditions can be used as guidance for design. Here, we propose two bimolecular implementations of qICMs both designed to reach adaptation of gene expression to fluctuations in cellular resources.