Antithetic integral feedback for the robust control of monostable and oscillatory biomolecular circuits

Abstract

Biomolecular feedback systems are now a central application area of interest within control theory. While classical control techniques provide invaluable insight into the function and design of both natural and synthetic biomolecular systems, there are certain aspects of biological control that have proven difficult to analyze with traditional methods. To this end, we describe here how the recently developed tools of dominance analysis can be used to gain insight into the nonlinear behavior of the antithetic integral feedback circuit, a recently discovered control architecture which implements integral control of arbitrary biomolecular processes using a simple feedback mechanism. We show that dominance theory can predict both monostability and periodic oscillations in the circuit, depending on the corresponding parameters and architecture. We then use the theory to characterize the robustness of the asymptotic behavior of the circuit in a nonlinear setting.

Compact attractors of an antithetic integral feedback system have a simple structure

“…and our assumptions on the sign pattern of the Jacobian imply that (J(x(t))) [2] is Metzler for all t ≥ 0. Thus (19) implies that (Φ(t)) (2) ≥ 0 for all t ≥ 0.…”

“…Remark 3. Our goal in computing J [2] was to prove that (19) is a cooperative dynamical system. We note in passing that under certain conditions J [2] can also be used to rule out the existence of limit cycles [18].…”

“…is a cone of rank k. This has been used to prove a Poincaré-Bendixson property for a certain type of dynamical systems in [28]. A recent paper [19] studied the same biological model studied as we do, using a a similar technique, but only for certain specific parameter values. Their analysis suggests that the model does not have a fixed C(P ) as an invariant set.…”

Guidelines for designing the antithetic feedback motif

“…In [38,39], the authors determined analytic conditions on the reaction rate parameters of the antithetic feedback network such that the linearized closed loop system is stable. Subsequent research also discussed the stability of the nonlinear dynamics of closed loop antithetic feedback network [44,45]. Without stability, the antithetic feedback control would not be able to track the reference signal; instead, it would oscillate indefinitely.…”

“…Indeed, in the previous synthetic biological implementations of antithetic feedback in E. coli and S. cerevisiae, the antithetic feedback controller's relative steady-state error (as normalized to the reference) has been measured to be 5-50% [28,31]. Another consideration is the stability of the closed-loop antithetic feedback system since theoretical studies have demonstrated that, depending on the reaction rate parameters, the antithetic controller can become unstable and periodic oscillations can arise [39,42,44,45]. Therefore, it is important to consider how the stability, robustness, and steady-state error (performance) of the antithetic feedback motif depend on its synthetic biological implementation and to study in depth how to tune these properties [28,46].…”