Robust Perfect Adaptation in Biomolecular Reaction Networks

For control in biomolecular systems, the most basic objective of maintaining a small error in a target variable, say the expression level of some protein, is often difficult due to the presence of both large uncertainty of every type and intrinsic limitations on the controllers implementation. This paper explores the limits of biochemically plausible controller design for the problem of robust perfect adaptation (RPA), biologists term for robust steady state tracking. It is well-known that for a large class of nonlinear systems, a system has RPA iff it has integral feedback control (IFC), which has been used extensively in real control systems to achieve RPA. However, we show that due to intrinsic physical limitations on the dynamics of chemical reaction networks (CRNs), cells cannot implement IFC directly in the concentration of a chemical species. This contrasts with electronic implementations, particularly digital, where it is trivial to implement IFC directly in a single state. Therefore, biomolecular systems have to achieve RPA by encoding the integral control variable into the network architecture of a CRN. We describe a general framework to implement RPA in CRNs and show that well-known network motifs that achieve RPA, such as (negative) integral feedback (IFB) and incoherent feedforward (IFF), are examples of such implementations. We also develop methods to solve the problem of designing integral feedback variables for unknown plants. This standard control notion is surprisingly nontrivial and relatively unstudied in biomolecular control. The methods developed here connect different existing fields and approaches on the problem of biomolecular control, and hold promise for systematic chemical reaction controller synthesis as well as analysis.