How Much Spatial Control Is Enough? Subdomain Optimal Control of Reaction-Diffusion Systems in Synthetic Developmental Biology

Abstract

Reaction--diffusion systems can produce spatial patterns such as stripes and spots by diffusion-driven instability. In synthetic developmental biology, such patterns are engineered in cell populations, and steering them toward a prescribed target pattern can be formulated as an optimal control problem. When the control acts on the entire spatial domain, the existence of optimal controls and the first-order optimality conditions are well established. In experiments, the control is applied with light by optogenetic tools, and the light illuminates a chosen region of the tissue, so the control acts on a subdomain. We consider the setting where the control is restricted to a subdomain fixed in advance, for a synthetic circuit in which the slowly diffusing activator Nodal and the fast-diffusing inhibitor Lefty form the pattern. We study the optimality loss, the difference between the optimal costs of the subdomain problem and of the full-domain problem, and we ask how much coverage keeps it within a desired precision. In simulations with one parameter set and two target patterns, nine squares give a smaller terminal error than one square at the same coverage. Moreover, the error depends on the average distance from the uncontrolled points to the control region. In particular, small errors occur when this distance is comparable to the length scale set by Nodal diffusion and degradation.