Biomolecular LQR under Partial Observation
Xiaoyu Zhang, Zhou Fang
TL;DR
This work introduces a biomolecular Linear Quadratic Regulator (LQR) framework to study gene regulatory networks and shows that LQR-designed controllers implemented via Hill-function motifs reproduce common network motifs like negative autoregulation and incoherent feedforward loops under partial observation. By deriving analytical mappings between LQR parameters and biomolecular reaction rates, the authors connect the quadratic cost weights $Q$ to environmental survival goals, defining performance bounds such as $Q<3ar{ ext{ } }^2$ that constrain feasible control in simple circuits. The approach is validated through one- and two-gene simulations, demonstrating that biomolecular LQR controllers can achieve faster regulation, better disturbance rejection, and lower cumulative cost compared to open-loop designs, even when observing only partial states. Overall, the paper provides a theoretical link between evolutionary pressures and regulatory motifs, offering a principled framework for designing robust synthetic biological circuits that balance performance and resource costs.
Abstract
This paper introduces a biomolecular Linear Quadratic Regulator (LQR) to investigate the design principles of gene regulatory networks. We show that for fundamental gene regulation network, the bio-controller derived from LQR theory precisely recapitulate natural network motifs, such as auto-regulation and incoherent feedforward loops. This emulation arises from a fundamental principle: the LQR cost function mathematically encodes environmental survival demands, which subsequently drives the selection of both network topology and biochemical parameters. Our work thus establishes a theoretical basis for interpreting biological circuit design, directly linking evolutionary pressures to observable regulatory structures.