Leveraging network structure in nonlinear control

Abstract

Over the last twenty years, dynamic modeling of biomolecular networks has exploded in popularity. Many of the classical tools for understanding dynamical systems are unwieldy in the highly nonlinear, poorly constrained, high-dimensional systems that often arise from these modeling efforts. Understanding complex biological systems is greatly facilitated by purpose-built methods that leverage common features of such models, such as local monotonicity, interaction graph sparsity, and sigmoidal kinetics. Here, we review methods for controlling the systems of ordinary differential equations used to model biomolecular networks. We focus on methods that make use of the structure of the network of interactions to help inform, which variables to target for control, and highlight the computational and experimental advantages of such approaches. We also discuss the importance of nonperturbative methods in biomedical and experimental molecular biology applications, where finely tuned interventions can be difficult to implement. It is well known that feedback loops, and positive feedback loops in particular, play a major determining role in the dynamics of biomolecular networks. In many of the methods we cover here, control over system trajectories is realized by overriding the behavior of key feedback loops.

Fig. 1. Network structure and minimal feedback vertex sets of Eqs. 2 and 3.

In this example, Eq. 2 is written in a privileged, or natural set of coordinates that correspond to the activities of individual biomolecules. The minimal feedback vertex set of Eq. 2 is unique, and consists of variables Y and Z, highlighted in blue in the network on the left (panel a); in addition, all feedback vertex sets contain this as a subset due to the self-regulation of these two variables. This implies that overriding Y and Z is sufficient to attain either of the system’s two attractors. Furthermore, in this example, this control set is minimal. A change of variables (depicted in panel b) gives rise to a system with a smaller feedback vertex set, consisting only of a single variable, $y=\left(Y+Z\right)/3$, highlighted in green. However, in order to achieve the necessary override of y in the laboratory, Y and Z must both be manipulated. This example illustrates the importance of the natural coordinates of the system when designing control interventions.

Fig. 2. Schematic of stable motif procedure applied to the system of Eq. 6.

The network structure of the system is depicted in panel a. Circle-tipped arrows represent inhibition, while wedge-tipped arrows represent activation. In panel b, the bounding systems in Eqs. 8 and 9 are depicted alongside the original system. Note that these bounding systems are monotone input/output systems. Panel c illustrates the characteristic feedback method of identifying steady states in monotone input/output systems, wherein the effect of overriding a feedback vertex set (FVS), in this case, the set containing $y_m$ or $y_M$, is considered, as described in the example of Eq. 4.

Fig. 3. Phase portraits and nullclines of the bounding systems of Eqs. 8 and 9.

The phase portrait of the bounding $x_M,y_M$ system is depicted in the top left. The trajectories of the x,y system are such that $x\left(t\right)\ge x_M\left(t\right)$ and $y\left(t\right)\le y_M\left(t\right)$. The phase portrait of the $x_m,y_m$ system is depicted in the top right and bottom left corners for two values of the parameter ϵ. For $ϵ=0.75$ (weak control strength), the system exhibits a steady state with small x and large y. Because $x_m\left(t\right)\ge x\left(t\right)\ge x_M\left(t\right)$ and $y_m\left(t\right)\le y\left(t\right)\le y_M\left(t\right)$ both hold, this steady state implies the existence of a subset of the state space that cannot be escaped by varying $u\left(t\right)$ in Eq. 6; this region is highlighted in red in the bottom right figure. Note that in the strong control case ($ϵ=0.25$), there is no such steady state in the bounding system, and thus the control is able to drive the system from a low-x, high-y state to a high-x, low-y state.