Geometry-induced protein pattern formation

Significance Biological cells need the ability to guide intracellular processes to specific spatial locations. This requires biochemical processes to sense and adapt to the geometry of the organism. Previously suggested mechanisms either assume proteins that are able to directly sense membrane curvature or are based on nonlinear diffusion–reaction systems that can generate geometry-adapted patterns. The latter, however, requires fine-tuning of the reaction rates. Here, we show that geometry adaption already follows from generic chemical dynamics. We present a simple reaction module based on generic reactions that establishes geometry-dependent patterns robustly without the need to tune kinetic rates nor any explicit curvature-sensing mechanism. Protein patterns are known to adapt to cell shape and serve as spatial templates that choreograph downstream processes like cell polarity or cell division. However, how can pattern-forming proteins sense and respond to the geometry of a cell, and what mechanistic principles underlie pattern formation? Current models invoke mechanisms based on dynamic instabilities arising from nonlinear interactions between proteins but neglect the influence of the spatial geometry itself. Here, we show that patterns can emerge as a direct result of adaptation to cell geometry, in the absence of dynamical instability. We present a generic reaction module that allows protein densities robustly to adapt to the symmetry of the spatial geometry. The key component is an NTPase protein that cycles between nucleotide-dependent membrane-bound and cytosolic states. For elongated cells, we find that the protein dynamics generically leads to a bipolar pattern, which vanishes as the geometry becomes spherically symmetrical. We show that such a reaction module facilitates universal adaptation to cell geometry by sensing the local ratio of membrane area to cytosolic volume. This sensing mechanism is controlled by the membrane affinities of the different states. We apply the theory to explain AtMinD bipolar patterns in Δ EcMinDE Escherichia coli. Due to its generic nature, the mechanism could also serve as a hitherto-unrecognized spatial template in many other bacterial systems. Moreover, the robustness of the mechanism enables self-organized optimization of protein patterns by evolutionary processes. Finally, the proposed module can be used to establish geometry-sensitive protein gradients in synthetic biological systems.