Limits to positional information in boundary-driven systems

Chemical gradients can be used by a particle to determine its position. This \textit{positional information} is of crucial importance, for example in developmental biology in the formation of patterns in an embryo. The central goal of this paper is to study the fundamental physical limits on how much positional information can be stored inside a system. To achieve this, we study positional information for general boundary-driven systems, and derive, in the near-equilibrium regime, a universal expression involving only the chemical potential and density gradients of the system. We also conjecture that this expression serves as an upper bound on the positional information of boundary driven systems beyond linear response. To support this claim, we test it on a broad range of solvable boundary-driven systems.