Geometry-induced pattern formation

Proteins play a critical role in organizing a multitude of cellular process, including cell division, cell motility, and nutrient uptake. To achieve this, proteins form spatiotemporal patterns that are governed by an interplay between transport processes and local biochemical reactions. Notably, protein patterns often emerge along the cell membrane, which is a dynamic object as cells alter their shape during various processes. This can lead to an intricate feedback loop between protein patterns and dynamic shape changes of the membrane, for which it is difficult to extract the mechanisms underlying the dynamics. We will discuss a conceptual model for cell polarity on a time-evolving geometry [1]. Utilizing tools from differential geometry, we derive the equations describing mass-conserving reaction-diffusion systems on dynamic membranes. Based on a recently developed framework for mass-conserving reaction-diffusion systems [2], we then analytically derive a criterion that links the onset of pattern formation to the phase-space structure of the reaction-diffusion system. We show that shape deformations can regionally induce, suppress, and spatially shift pre-existing patterns. Moreover, we demonstrate that the feedback loop between membrane shape and reaction-diffusion dynamics leads to traveling wave and standing wave patterns that do not occur on static membrane geometries. Our work underscores the local conformation of the geometry and the total protein mass as the relevant dynamic variables for pattern formation.

[1] Geometry-induced patterns through mechanochemical coupling, L. Würthner, A. Goychuk, and E. Frey. arXiv:2205.02820 (2022)

[2] Phase-space geometry of mass-conserving reaction-diffusion dynamics, F. Brauns, J. Halatek, and E. Frey. Phys. Rev. X 10, 041036 (2020)