How can proteins take derivatives?

Many cellular processes, such as cell division and cell motility, rely crucially on the dynamic localization of proteins in space and time. These localization patterns emerge collectively from local molecular interactions of proteins. To analyze how the interplay of diffusion and protein interactions on a nanometer scale influence the protein patterns on the cellular scale, the framework of reaction-diffusion models has proven useful. The study of such systems goes back to Turing, who showed how patterns can emerge from a homogenous steady state when two reactive components have different diffusivities. However, in nature, systems typically develop in a heterogeneous and temporally evolving environment and from one pattern into another, rather than from a homogeneous steady state into a pattern.

Here, we study how protein localization patterns arise in heterogeneous systems. We show how localization patterns of upstream regulators dynamically control pattern formation. In particular, we identify concrete mechanisms through which downstream systems can effectively take spatial or temporal “derivatives” of the upstream protein concentration profile. Such mechanisms allow the cell to precisely and robustly control spatial protein organization.