Structural Stability Properties of Antithetic Integral (Rein) Control with Output Inhibition

TL;DR

The paper develops a comprehensive control-theoretic framework for the Antithetic Integral Rein Controller (AIRC) with output inhibition, focusing on robust structural stability and perfect adaptation across unimolecular and nonlinear biochemical networks. It reveals that the niAIC with output inhibition behaves as a filtered PI controller, enabling access to a broad set of admissible set-points and improved transient behavior, including switching between naAIC-like and niAIC-like regimes under strong sequestration. By leveraging positive-real transfer functions, Metzler-Hurwitz stability, and passivity-based interconnections, the authors derive conditions under which the closed-loop system remains locally exponentially stable for all positive controller gains, including cases where the underlying network is output unstable. They provide practical computational tools (LMIs, Nyquist/SPR tests) to verify admissible set-points and stability, and extend results to cooperative and Michaelis-Menten nonlinear networks, with an intein-based implementation proposal. Overall, the work advances robust, structure-driven control of reaction networks, offering guidelines for designing controllers that maintain stability and perfect adaptation despite strong parameter uncertainties.

Abstract

Perfect adaptation is a well-studied biochemical homeostatic behavior lying at the core of biochemical regulation. While the concepts of homeostasis and perfect adaptation are not new, their underlying mechanisms and associated biochemical regulation motifs are not yet fully understood. Insights from control theory unraveled the connections between perfect adaptation and integral control, a prevalent engineering control strategy. In particular, the recently introduced Antithetic Integral Controller (AIC) has been shown to successfully ensure perfect adaptation properties to the network it is connected to. The complementary structure of the two molecules the AIC relies upon allows for a versatile way to control biochemical networks, a property which gave rise to an important body of literature pertaining to mathematically elucidating its properties, generalizing its structure, and developing experimental methods for its implementation. The Antithetic Integral Rein Controller (AIRC), an extension of the AIC in which both controller molecules are used for control, holds many promises as it supposedly overcomes certain limitations of the AIC. We focus here on an AIRC structure with output inhibition that combines two AICs in a single structure. We demonstrate that rhis controller ensure structural stability and structural perfect adaptation properties for the controlled network under mild assumptions, meaning that this property is independent of the parameters of the network and the controller. The results are very general and valid for the class of unimolecular mass-action networks as well as more general networks, including cooperative and Michaelis-Menten networks. We also provide a systematic and accessible computational way for verifying whether a given network satisfies the conditions under which the structural property would hold.

Figures (22)

• Figure 1: (a). Different possible structures of the AIRC depending on whether the controller species $\boldsymbol{Z_{1}}$ and $\boldsymbol{Z_{2}}$ act as activators or inhibitors on the actuated species $\boldsymbol{X_{1}}$ and $\boldsymbol{X_{m}}$ and whether the actuated species act as activators or inhibitors for the controlled species $\boldsymbol{X_{n}}$. In the first column, $\boldsymbol{X_{1}}$ is an activator for $\boldsymbol{X_{n}}$ whereas it is an inhibitor in the second one. Similarly, in the first row $\boldsymbol{X_{m}}$ is an inhibitor of $\boldsymbol{X_{n}}$ where it is an activator in the second one. The roles of the controller species is chosen so that the overall feedback loop is a negative feedback loop and have opposite action on the controlled species. For instance, when $\boldsymbol{X_{1}}$ activates $\boldsymbol{X_{n}}$ and $\boldsymbol{X_{m}}$ inhibits it, a suitable AIRC is given by the aaAIRC, since that $\boldsymbol{X_{n}} \boldsymbol{Z_{2}} \boldsymbol{Z_{1}} \boldsymbol{X_{1}} \boldsymbol{X_{n}}$ and $\boldsymbol{X_{n}} \boldsymbol{Z_{2}} \boldsymbol{X_{m}} \boldsymbol{X_{n}}$, showing that $\boldsymbol{Z_{1}}$ acts as an activator and $\boldsymbol{Z_{2}}$ as a repressor, as required to make the topology a rein controller. One can also observe that the two feedback loops implemented by the aaAIRC are negative feedback loops. Analogous conclusions hold for all the topologies. (b) The aiAIRC with output inhibition consists of two AICs: an naAIC that activates the production of $\boldsymbol{X_{1}}$ (blue pointed arrow) and an niAIC that directly inhibits the output $\boldsymbol{X_{n}}$ (red blunt head arrow).
• Figure 2: Left. Spectral abscissa (i.e. real-part of the rightmost eigenvalue) of the system associated with the \ref{['main:eq:RN:maturation']}, \ref{['main:eq:AIC:niAICoi']} with the parameters $\gamma_1=1$, $\gamma_2=\gamma_3=2$, $k_{21}=1$, $k_{32}=2$, $k_0=10$, $k_{13}=1$, $\nu=0$, $\eta=100/k_p$, and $\theta=1$ and for various values for $\mu$ and $k_p$. We have that $g_0=10$ and calculations show that the spectral abscissa of $A$ is $-0.3044$ while the spectral abscissa of the closed-loop system may reach smaller values, which indicates that this controller is able to improve the convergence properties of the system near the equilibrium point. Right. Time domain evolution of the concentrations of $\boldsymbol{X_{3}}$ for various values for the set-point $\mu/\theta$ and controller gains $k_p$. The left column depicts simulation results for zero initial conditions (convergence properties) whereas the right column depicts the response of the closed-loop network when the parameter $k_{32}$ changes from 2 to 3 at $t=5$ (perfect adaptation property).
• Figure 3: A. Spectral abscissa (i.e. real-part of the rightmost eigenvalue) of the system associated with the \ref{['main:eq:RN:maturation']}, \ref{['main:eq:AIC:niAICoi']} with the parameters $\gamma_1=1$, $\gamma_2=\gamma_3=2$, $k_{21}=1$, $k_{32}=2$, $k_0=10$, $k_{13}=3$, $\nu=0$, $\eta=100/k_p$, and $\theta=1$ and for various values for $\mu$ and $k_p$. We have that $g_0=-10$, $g_n=-1$, and calculations show that the spectral abscissa of $A$ is $0.2188$ while the spectral abscissa of the closed-loop system may reach smaller values, which indicates that this controller is able to stabilize the network and improve the convergence properties of the system near the equilibrium point. B. Time domain evolution of the concentrations of $\boldsymbol{X_{3}}$ for various values for the set-point $\mu/\theta$ and controller gains $k_p$. The left column depicts simulation results for zero initial conditions (convergence properties) whereas the right column depicts the response of the closed-loop network when the parameter $k_{32}$ changes from 2 to 3 at $t=5$ (perfect adaptation property).
• Figure 4: A. Spectral abscissa (i.e. real-part of the rightmost eigenvalue) of the system associated with the \ref{['main:eq:RN:maturation']}, \ref{['main:eq:AIC:niAICoi']} with the parameters $\gamma_1=1$, $\gamma_2=\gamma_3=2$, $k_{21}=1$, $k_{32}=2$, $k_0=10$, $k_{13}=1$, $\nu=3$, $\eta=100/k_p$, and $\theta=1$ and for various values for $\mu$ and $k_p$. We have that $g_0=-5$, $g_n=-1/2$, and calculations show that the spectral abscissa of $A$ is $1.2695$ while the spectral abscissa of the closed-loop system may reach smaller values, which indicates that this controller is able to stabilize the network and improve the convergence properties of the system near the equilibrium point. B. Time domain evolution of the concentrations of $\boldsymbol{X_{3}}$ for various values for the set-point $\mu/\theta$ and controller gains $k_p$. The left column depicts simulation results for zero initial conditions (convergence properties) whereas the right column depicts the response of the closed-loop network when the parameter $k_{32}$ changes from 2 to 3 at $t=5$ (perfect adaptation property).
• Figure 5: Stability regions in the $(k_p, k_i)$-plane for the unique equilibrium point of system \ref{['main:eq:mainsystCL']}, associated with the gene expression network with protein maturation \ref{['main:eq:RN:maturation']} controlled by the aiAIRC with output inhibition \ref{['main:eq:AIC:aiAIRCoi']}. Common parameters for all simulation panels are: $\gamma_1 = 1$; $\gamma_2 = \gamma_3 = 2$; $k_{21} = 1$; $k_{32} = 1$; $k_{13} = 0$; $\nu = 0$; $\eta = 100$; $\theta = 1$; and $\mu = 2.5$. In the first panel, $k_0 = 5$ (which gives $g_0 = 1.25$), so $r > g_0$; in the second panel, $k_0 = 10$ (yielding $g_0 = 2.5$), so $r = g_0$; and in the third panel, $k_0 = 15$ (resulting in $g_0 = 3.75$), so $r < g_0$. The red regions along the boundary of the nonnegative orthant represent areas where the system either lacks a nonnegative equilibrium point (i.e., the set-point is not admissible) or where the unique nonnegative equilibrium point exists but is unstable.

Theorems & Definitions (64)

• Definition 1.1: Perfect Adaptation
• Definition 1.2: Perfect Adaptation Design Problem
• Definition 1.3: Set-point admissibility
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 2.4
• Proposition 2.5
• Theorem 2.6
• Theorem 2.7