  Shapes of biological membranes are dynamically regulated in living cells.
Although membrane shape deformation by proteins at thermal equilibrium has been
extensively studied, nonequilibrium dynamics have been much less explored.
Recently, chemical reaction propagation has been experimentally observed in
plasma membranes. Thus, it is important to understand how the
reaction-diffusion dynamics are modified on deformable curved membranes. Here,
we investigated nonequilibrium pattern formation on vesicles induced by
mechanochemical feedback between membrane deformation and chemical reactions,
using dynamically triangulated membrane simulations combined with the
Brusselator model. We found that membrane deformation changes stable patterns
relative to those that occur on a non-deformable curved surface, as determined
by linear stability analysis. We further found that budding and multi-spindle
shapes are induced by Turing patterns, and we also observed the transition from
oscillation patterns to stable spot patterns. Our results demonstrate the
importance of mechanochemical feedback in pattern formation on deforming
membranes.
